diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index 0250b21d..e8abf785 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -11,6 +11,7 @@ repos:
   - id: check-merge-conflict
   - id: check-xml
   - id: check-yaml
+    args: ['--unsafe']
   - id: debug-statements
   - id: end-of-file-fixer
   - id: requirements-txt-fixer
diff --git a/README.md b/README.md
index bfded412..a88f7b49 100644
--- a/README.md
+++ b/README.md
@@ -1,27 +1,120 @@
-<!-- These are examples of badges you might want to add to your README:
-     please update the URLs accordingly
+# foapy
 
-[![Built Status](https://api.cirrus-ci.com/github/<USER>/foapy.svg?branch=main)](https://cirrus-ci.com/github/<USER>/foapy)
-[![ReadTheDocs](https://readthedocs.org/projects/foapy/badge/?version=latest)](https://foapy.readthedocs.io/en/stable/)
-[![Coveralls](https://img.shields.io/coveralls/github/<USER>/foapy/main.svg)](https://coveralls.io/r/<USER>/foapy)
-[![PyPI-Server](https://img.shields.io/pypi/v/foapy.svg)](https://pypi.org/project/foapy/)
-[![Conda-Forge](https://img.shields.io/conda/vn/conda-forge/foapy.svg)](https://anaconda.org/conda-forge/foapy)
-[![Monthly Downloads](https://pepy.tech/badge/foapy/month)](https://pepy.tech/project/foapy)
-[![Twitter](https://img.shields.io/twitter/url/http/shields.io.svg?style=social&label=Twitter)](https://twitter.com/foapy)
--->
+Python library for Formal Order Analysis (FOA) of symbolic sequences.
 
-[![Project generated with PyScaffold](https://img.shields.io/badge/-PyScaffold-005CA0?logo=pyscaffold)](https://pyscaffold.org/)
+## What is FOA?
 
-# foapy
+Formal Order Analysis (FOA) studies the structure of symbolic sequences by:
+
+- Mapping a sequence to its order and alphabet (the set of unique symbols in order of first appearance).
+- Extracting intervals between repeated occurrences of the same symbol under different boundary modes.
+- Quantifying structure with characteristics (e.g., arithmetic/geometric means, volume, average remoteness, depth, descriptive/identifying information, regularity, uniformity).
+
+This enables robust analysis for text, biological sequences, musical motifs, logs, and any categorical time series.
+
+## Features
+
+- **Alphabets and Orders:** Define alphabets and compute orders of sequences.
+- **Intervals and Chains:** Extract intervals for multiple binding and boundary modes.
+- **Congeneric Decomposition:** Decompose sequences into congeneric components.
+- **Mathematical Characteristics:** Arithmetic mean, geometric mean, volume, average remoteness, depth, descriptive and identifying information, regularity, uniformity.
+- **Masked-array Support:** FOA on partially observed sequences via `foapy.ma`.
+- **Extensive Documentation:** Theory, algorithms, and examples.
+
+## Installation
+
+- From PyPI:
+
+```bash
+pip install foapy
+```
+
+- From source (with tox workflows):
+
+```bash
+git clone https://github.com/intervals-mining-lab/foapy.git
+cd foapy
+python -m pip install --upgrade pip
+python -m pip install tox
+
+# Run tests (isolated env)
+tox -e default
+
+# Build distribution artifacts (sdist and wheel)
+tox -e build
+
+# Optionally clean build artifacts
+tox -e clean
+```
+
+## Quick start
+
+Compute order, intervals, and a characteristic:
+
+```python
+import foapy
+
+# 1) Order and alphabet
+source = ['a', 'b', 'a', 'c', 'd']
+order = foapy.order(source)
+print(order)  # [0 1 0 2 3]
+
+order_arr, alphabet = foapy.order(source, True)
+print(order_arr, alphabet)  # [0 1 0 2 3] ['a' 'b' 'c' 'd']
+
+# 2) Intervals (binding to start, normal mode)
+from foapy import binding, mode
+intervals = foapy.intervals(['a', 'b', 'a', 'c', 'a', 'd'], binding.start, mode.normal)
+print(intervals)  # [1 2 2 3 2 5]
+
+# 3) A characteristic (volume = product of intervals)
+val = foapy.characteristics.volume(intervals)
+print(val)  # 192
+```
+
+Masked arrays (optional):
+
+```python
+import numpy as np
+import numpy.ma as ma
+import foapy
+
+seq = ma.masked_array(['a', 'b', 'a', 'c', 'd'], mask=[0, 1, 0, 0, 0])
+order_ma = foapy.ma.order(seq)
+intervals_grouped = foapy.ma.intervals(order_ma, foapy.binding.start, foapy.mode.normal)
+u = foapy.characteristics.uniformity(intervals_grouped)
+print(u)
+```
+
+## Project Structure
+
+- **Source Code:** [`./src`](./src)
+- **Documentation:** [`./docs`](./docs)
+- **Tests:** [`./tests`](./tests)
+
+## Documentation
+
+Online documentation: [intervals-mining-lab.github.io/foapy](https://intervals-mining-lab.github.io/foapy).
+
+The documentation in [`./docs`](./docs) covers fundamentals, algorithms, characteristics, applications, and development notes.
+
+Build and serve the docs locally (via tox):
 
-> Formal order analysis library
+```bash
+# Build docs into docs/_build
+tox -e docs
 
-A longer description of your project goes here...
+# Serve docs with live-reload for development
+tox -e docsserve
+```
 
+## Testing
 
-<!-- pyscaffold-notes -->
+Run the test suite (via tox):
 
-## Note
+```bash
+tox -e default
 
-This project has been set up using PyScaffold 4.5. For details and usage
-information on PyScaffold see https://pyscaffold.org/.
+# Pass arguments to pytest after --, e.g. run a subset:
+tox -e default -- -k order -q
+```
diff --git a/docs/assets/js/mathjax.js b/docs/assets/js/mathjax.js
index 0c7803cf..561944cd 100644
--- a/docs/assets/js/mathjax.js
+++ b/docs/assets/js/mathjax.js
@@ -1,9 +1,11 @@
 window.MathJax = {
+    loader: {load: ['[tex]/colortbl']},
     tex: {
       inlineMath: [["\\(", "\\)"]],
       displayMath: [["\\[", "\\]"]],
       processEscapes: true,
-      processEnvironments: true
+      processEnvironments: true,
+      packages: {'[+]': ['colortbl']},
     },
     options: {
       ignoreHtmlClass: ".*|",
diff --git a/docs/fundamentals/characteristics/arithmetic_mean.md b/docs/fundamentals/characteristics/arithmetic_mean.md
deleted file mode 100644
index 3f3ed90f..00000000
--- a/docs/fundamentals/characteristics/arithmetic_mean.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Arithmetic Mean
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/average_remoteness.md b/docs/fundamentals/characteristics/average_remoteness.md
deleted file mode 100644
index 415f7290..00000000
--- a/docs/fundamentals/characteristics/average_remoteness.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Average Remoteness
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/depth.md b/docs/fundamentals/characteristics/depth.md
deleted file mode 100644
index 16f82bad..00000000
--- a/docs/fundamentals/characteristics/depth.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Depth
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/descriptive_information.md b/docs/fundamentals/characteristics/descriptive_information.md
deleted file mode 100644
index 57d65081..00000000
--- a/docs/fundamentals/characteristics/descriptive_information.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Descriptive Information
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/geometric_mean.md b/docs/fundamentals/characteristics/geometric_mean.md
deleted file mode 100644
index bfe6dcdb..00000000
--- a/docs/fundamentals/characteristics/geometric_mean.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Geometric Mean
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/identifying_information.md b/docs/fundamentals/characteristics/identifying_information.md
deleted file mode 100644
index 6bd90c57..00000000
--- a/docs/fundamentals/characteristics/identifying_information.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Identifying Information
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/index.md b/docs/fundamentals/characteristics/index.md
deleted file mode 100644
index ba1feea7..00000000
--- a/docs/fundamentals/characteristics/index.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Characteristics
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/periodicity.md b/docs/fundamentals/characteristics/periodicity.md
deleted file mode 100644
index a26aaa16..00000000
--- a/docs/fundamentals/characteristics/periodicity.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Periodicity
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/uniformity.md b/docs/fundamentals/characteristics/uniformity.md
deleted file mode 100644
index 35fc8f22..00000000
--- a/docs/fundamentals/characteristics/uniformity.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Uniformity
-
-Coming soon
diff --git a/docs/fundamentals/characteristics/volume.md b/docs/fundamentals/characteristics/volume.md
deleted file mode 100644
index 0cee8e6a..00000000
--- a/docs/fundamentals/characteristics/volume.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Volume
-
-Coming soon
diff --git a/docs/fundamentals/congeneric_decomposition/alphabet.md b/docs/fundamentals/congeneric_decomposition/alphabet.md
new file mode 100644
index 00000000..b1f21ba9
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/alphabet.md
@@ -0,0 +1,357 @@
+# Alphabet of Congeneric sequences
+
+_Alphabet_ of [_Congeneric sequences_](./sequences.md#mathematical-definition) is an [_Alphabet_](../order/alphabet.md#mathematical-definition).
+
+
+=== "Congeneric sequences"
+    ``` mermaid
+    block-beta
+    columns 31
+
+    space:2 i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+    i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+    i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+    i28["28"] i29["29"]
+
+    space j1["1"] s1["I"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["I"] s8["-"] s9["-"] s10["-"]
+    s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+    s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"]
+
+    space j2["2"] n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+    n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+    n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"]
+
+    space j3["3"] t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+    t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+    t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"]
+
+    space j4["4"] e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+    e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["-"] e20["-"]
+    e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"]
+
+    space j5["5"] l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+    l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+    l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"]
+
+    space j6["6"] g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+    g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+    g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"]
+
+    space j7["7"] c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+    c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+    c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"]
+
+    space j8["8"] sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+    sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+    sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28["-"] sp29["-"]
+
+    space j9["9"] b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+    b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+    b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#ffffff
+    classDef index fill:#ffffff,stroke-width:0px
+
+    class s2,s3,s4,s5,s6,s8,s9,s10 index
+    class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 index
+    class s21,s22,s23,s24,s25,s26,s27,s28,s29 index
+
+    class n1,n3,n4,n5,n6,n7,n8,n9 index
+    class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 index
+    class n21,n22,n23,n24,n25,n26,n27,n28,n29 index
+
+    class t1,t2,t4,t5,t6,t7,t8,t9,t10 index
+    class t11,t12,t13,t14,t15,t16,t18,t19,t20 index
+    class t21,t22,t23,t24,t25,t27,t28 index
+
+    class e1,e2,e3,e5,e6,e7,e8,e10 index
+    class e11,e13,e14,e15,e16,e17,e18,e19,e20 index
+    class e21,e22,e23,e24,e25,e26,e27,e28,e29 index
+
+    class l1,l2,l3,l4,l7,l8,l9,l10 index
+    class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 index
+    class l21,l22,l23,l25,l26,l27,l28,l29 index
+
+    class g1,g2,g3,g4,g5,g6,g7,g9,g10 index
+    class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 index
+    class g21,g22,g23,g24,g25,g26,g27,g28,g29 index
+
+    class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 index
+    class c12,c13,c14,c15,c16,c17,c18,c19,c20 index
+    class c21,c22,c23,c24,c25,c26,c27,c28,c29 index
+
+    class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 index
+    class sp11,sp12,sp14,sp15,sp17,sp18,sp19 index
+    class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29 index
+
+    class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 index
+    class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 index
+    class b21,b23,b24,b25,b26,b27,b28,b29 index
+
+
+    class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+    class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+    class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+
+    class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+    ```
+
+
+=== "Alphabet and Congeneric sequences"
+    ``` mermaid
+    block-beta
+    columns 31
+
+    space:2 i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+    i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+    i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+    i28["28"] i29["29"]
+
+    a1["I"] j1["1"] s1["I"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["I"] s8["-"] s9["-"] s10["-"]
+    s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+    s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"]
+
+    a2["N"] j2["2"] n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+    n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+    n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"]
+
+    a3["T"] j3["3"] t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+    t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+    t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"]
+
+    a4["E"] j4["4"] e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+    e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["-"] e20["-"]
+    e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"]
+
+    a5["L"] j5["5"] l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+    l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+    l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"]
+
+    a6["G"] j6["6"] g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+    g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+    g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"]
+
+    a7["C"] j7["7"] c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+    c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+    c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"]
+
+    a8[" "] j8["8"] sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+    sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+    sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28["-"] sp29["-"]
+
+    a9["B"] j9["9"] b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+    b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+    b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#fff
+    classDef index fill:#fff,stroke-width:0px
+    classDef hide fill:#fff,color:#fff,stroke-width:0px
+
+    class s2,s3,s4,s5,s6,s8,s9,s10 index
+    class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 index
+    class s21,s22,s23,s24,s25,s26,s27,s28,s29 index
+
+    class n1,n3,n4,n5,n6,n7,n8,n9 index
+    class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 index
+    class n21,n22,n23,n24,n25,n26,n27,n28,n29 index
+
+    class t1,t2,t4,t5,t6,t7,t8,t9,t10 index
+    class t11,t12,t13,t14,t15,t16,t18,t19,t20 index
+    class t21,t22,t23,t24,t25,t27,t28 index
+
+    class e1,e2,e3,e5,e6,e7,e8,e10 index
+    class e11,e13,e14,e15,e16,e17,e18,e19,e20 index
+    class e21,e22,e23,e24,e25,e26,e27,e28,e29 index
+
+    class l1,l2,l3,l4,l7,l8,l9,l10 index
+    class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 index
+    class l21,l22,l23,l25,l26,l27,l28,l29 index
+
+    class g1,g2,g3,g4,g5,g6,g7,g9,g10 index
+    class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 index
+    class g21,g22,g23,g24,g25,g26,g27,g28,g29 index
+
+    class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 index
+    class c12,c13,c14,c15,c16,c17,c18,c19,c20 index
+    class c21,c22,c23,c24,c25,c26,c27,c28,c29 index
+
+    class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 index
+    class sp11,sp12,sp14,sp15,sp17,sp18,sp19 index
+    class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29 index
+
+    class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 index
+    class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 index
+    class b21,b23,b24,b25,b26,b27,b28,b29 index
+
+
+    class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+    class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+    class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+
+    class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+    ```
+
+
+=== "Alphabet"
+
+    ``` mermaid
+    block-beta
+    columns 31
+
+    space:2 i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+    i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+    i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+    i28["28"] i29["29"]
+
+    a1["I"] j1["1"] s1["I"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["I"] s8["-"] s9["-"] s10["-"]
+    s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+    s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"]
+
+    a2["N"] j2["2"] n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+    n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+    n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"]
+
+    a3["T"] j3["3"] t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+    t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+    t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"]
+
+    a4["E"] j4["4"] e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+    e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["-"] e20["-"]
+    e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"]
+
+    a5["L"] j5["5"] l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+    l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+    l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"]
+
+    a6["G"] j6["6"] g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+    g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+    g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"]
+
+    a7["C"] j7["7"] c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+    c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+    c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"]
+
+    a8[" "] j8["8"] sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+    sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+    sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28["-"] sp29["-"]
+
+    a9["B"] j9["9"] b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+    b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+    b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#fff
+    classDef index fill:#fff,stroke-width:0px
+    classDef hide fill:#fff,color:#fff,stroke-width:0px
+
+    class s1,s2,s3,s4,s5,s6,s7,s8,s9,s10 hide
+    class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 hide
+    class s21,s22,s23,s24,s25,s26,s27,s28,s29 hide
+
+    class n1,n2,n3,n4,n5,n6,n7,n8,n9,n10 hide
+    class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 hide
+    class n21,n22,n23,n24,n25,n26,n27,n28,n29 hide
+
+    class t1,t2,t3,t4,t5,t6,t7,t8,t9,t10 hide
+    class t11,t12,t13,t14,t15,t16,t17,t18,t19,t20 hide
+    class t21,t22,t23,t24,t25,t26,t27,t28,t29 hide
+
+    class e1,e2,e3,e4,e5,e6,e7,e8,e9,e10 hide
+    class e11,e12,e13,e14,e15,e16,e17,e18,e19,e20 hide
+    class e21,e22,e23,e24,e25,e26,e27,e28,e29 hide
+
+    class l1,l2,l3,l4,l5,l6,l7,l8,l9,l10 hide
+    class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 hide
+    class l21,l22,l23,l24,l25,l26,l27,l28,l29 hide
+
+    class g1,g2,g3,g4,g5,g6,g7,g8,g9,g10 hide
+    class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 hide
+    class g21,g22,g23,g24,g25,g26,g27,g28,g29 hide
+
+    class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 hide
+    class c11,c12,c13,c14,c15,c16,c17,c18,c19,c20 hide
+    class c21,c22,c23,c24,c25,c26,c27,c28,c29 hide
+
+    class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 hide
+    class sp11,sp12,sp13,sp14,sp15,sp16,sp17,sp18,sp19,sp20 hide
+    class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29 hide
+
+    class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 hide
+    class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 hide
+    class b21,b22,b23,b24,b25,b26,b27,b28,b29 hide
+
+
+    class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 hide
+    class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 hide
+    class i21,i22,i23,i24,i25,i26,i27,i28,i29 hide
+
+    class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+    ```
+
+
+## Mathematical Definition
+
+Let $X_{-}$ is [_Partial Carrier set_](../partials_and_congenerics/carrier_set.md#mathematical-definition)
+
+Let $CS$ is [_Congeneric Sequences_](./sequences.md#as-m-tuple-of-congeneric-sequences)
+
+Let $alphabet_p$ is [_Alphabet of partial sequence_](../partials_and_congenerics/alphabet/index.md#mathematical-definition)
+
+Because by definition of _Congeneric sequences_ $\forall j \in \{1,...,m\}$ $CS(j)$ is [_Congeneric sequence_](../partials_and_congenerics/sequence/congeneric.md), so
+$|alphabet_p(CS(j))| = 1$ because of [_Alphabet of Congeneric sequences_](../partials_and_congenerics/alphabet/congeneric.md#mathematical-definition) lemma.
+
+Define _Alphabet of congeneric sequences_ function
+
+$$alphabet_c : \big\{\{1,...,m\} \times \{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\},$$
+
+$$alphabet_c(CS)(j) = alphabet_p(CS(j))(1)$$
diff --git a/docs/fundamentals/congeneric_decomposition/characteristics/descriptive_information.md b/docs/fundamentals/congeneric_decomposition/characteristics/descriptive_information.md
new file mode 100644
index 00000000..b747c47f
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/characteristics/descriptive_information.md
@@ -0,0 +1,57 @@
+# Descriptive Information
+
+The Descriptive Information
+
+## Mathematical Definition
+
+The descriptive information can be calculated
+
+
+### from Congeneric Intervals Chains
+
+Let $CIC$ is [_Congenerics Intervals Chains_](../intervals_chains.md#as-matrix-ml) defined as matrix
+
+$$
+CIC =
+\begin{pmatrix}
+\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
+\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
+\end{pmatrix}
+$$
+
+$$D=\prod_{j=1}^{m}{\left(\frac{1}{n_j} \times \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}\right)^{\frac{n_j}{n}}}$$
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    1, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}$ is count of _non-empty_ elements in $j$ _congeneric intervals chain_,
+$\Delta_{i,j}$ the $i$-th element of $j$-th _congeneric intervals chain_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
+
+
+### from Congenerics Intervals Distributions
+
+Let $CID$ is [_Congenerics Intervals Distributions_](../intervals_distribution.md#as-matrix-ml) defined as matrix
+
+$$
+CID =
+\begin{pmatrix}
+cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\
+cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l}
+\end{pmatrix}
+$$
+
+$$D=\prod_{j=1}^{m}{\left(\sum_{i=1}^{l}{\frac{i \times cid_{i,j}}{n_j}}\right)^{\frac{n_j}{n}}}$$
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{cid_{i,j}}$ is count of _non-empty_ elements in $j$ _congeneric intervals distribution_,
+$cid_{i,j}$ the $i$-th element of $j$-th _congeneric intervals distribution_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
diff --git a/docs/fundamentals/congeneric_decomposition/characteristics/identifying_information.md b/docs/fundamentals/congeneric_decomposition/characteristics/identifying_information.md
new file mode 100644
index 00000000..15ffe80d
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/characteristics/identifying_information.md
@@ -0,0 +1,58 @@
+# Identifying Information
+
+The Identifying Information or Entropy
+
+## Mathematical Definition
+
+The identifying information/entropy can be calculated
+
+
+### from Congeneric Intervals Chains
+
+Let $CIC$ is [_Congenerics Intervals Chains_](../intervals_chains.md#as-matrix-ml) defined as matrix
+
+$$
+CIC =
+\begin{pmatrix}
+\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
+\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
+\end{pmatrix}
+$$
+
+$$H=\frac {1} {n} * \sum_{j=1}^{m}{\Bigg(n_j \times \log_2 \bigg( \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}\bigg) - \log_2 n_j^{n_j} \Bigg)}$$
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    1, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}$ is count of _non-empty_ elements in $j$ _congeneric intervals chain_,
+$\Delta_{i,j}$ the $i$-th element of $j$-th _congeneric intervals chain_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
+
+
+### from Congenerics Intervals Distributions
+
+Let $CID$ is [_Congenerics Intervals Distributions_](../intervals_distribution.md#as-matrix-ml) defined as matrix
+
+$$
+CID =
+\begin{pmatrix}
+cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\
+cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l}
+\end{pmatrix}
+$$
+
+$$H=\frac {1} {n} * \sum_{j=1}^{m}{\Bigg(n_j \times \log_2 \sum_{i=1}^{l}{ \frac{i \times cid_{i,j}}{n_j}} \Bigg)}$$
+
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{cid_{i,j}}$ is count of _non-empty_ elements in $j$ _congeneric intervals distribution_,
+$cid_{i,j}$ the $i$-th element of $j$-th _congeneric intervals distribution_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
diff --git a/docs/fundamentals/congeneric_decomposition/characteristics/index.md b/docs/fundamentals/congeneric_decomposition/characteristics/index.md
new file mode 100644
index 00000000..e7d3122b
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/characteristics/index.md
@@ -0,0 +1,11 @@
+# Characteristics
+
+Applying any of [_interval-based characteristics_](../../order/characteristics/index.md) to each element of _Congenetric Itervals Distributions_ stack produce
+a vector of the characteristic. These vectors could be useful in analyzing parts of the sequence.
+
+In addition to that, there are four integral measures based on _congeneric intervals distributions_ that characterise the whole sequence
+
+- [Descriptive Information](./descriptive_information.md)
+- [Identifying Information](./identifying_information.md)
+- [Regularity](./regularity.md)
+- [Uniformity](./uniformity.md)
diff --git a/docs/fundamentals/congeneric_decomposition/characteristics/regularity.md b/docs/fundamentals/congeneric_decomposition/characteristics/regularity.md
new file mode 100644
index 00000000..64f4c65e
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/characteristics/regularity.md
@@ -0,0 +1,73 @@
+# Regularity
+
+The Regularity
+
+## Mathematical Definition
+
+The regularity can be calculated
+
+
+### from Congeneric Intervals Chains
+
+Let $CIC$ is [_Congenerics Intervals Chains_](../intervals_chains.md#as-matrix-ml) defined as matrix
+
+$$
+CIC =
+\begin{pmatrix}
+\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
+\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
+\end{pmatrix}
+$$
+
+$$r= \sqrt[n]{\prod_{j=1}^{m} \frac{
+\prod_{j=1}^{l} \Bigg\{\begin{array}{l}
+    \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
+    1, &   \Delta_{i,j} \in \{ - \}
+\end{array}}
+{{\left(\frac{1}{n_j} \times \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}\right)^{n_j}}
+}
+}$$
+
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    1, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}$ is count of _non-empty_ elements in $j$ _congeneric intervals chain_,
+$\Delta_{i,j}$ the $i$-th element of $j$-th _congeneric intervals chain_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
+
+
+### from Congenerics Intervals Distributions
+
+Let $CID$ is [_Congenerics Intervals Distributions_](../intervals_distribution.md#as-matrix-ml) defined as matrix
+
+$$
+CID =
+\begin{pmatrix}
+cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\
+cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l}
+\end{pmatrix}
+$$
+
+$$r= \sqrt[n]{\prod_{j=1}^{m} \frac{
+\prod_{j=1}^{l} \Bigg\{\begin{array}{l}
+    i \times cid_{i,j}, & cid_{i,j} \neq 0 \\
+    1, &   cid_{i,j} = 0
+\end{array}}
+{{\left(\frac{1}{n_j} \times \sum_{i=1}^{l}{(i \times cid_{i,j})}\right)^{n_j}}
+}
+}$$
+
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{cid_{i,j}}$ is count of _non-empty_ elements in $j$ _congeneric intervals distribution_,
+$cid_{i,j}$ the $i$-th element of $j$-th _congeneric intervals distribution_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
diff --git a/docs/fundamentals/congeneric_decomposition/characteristics/uniformity.md b/docs/fundamentals/congeneric_decomposition/characteristics/uniformity.md
new file mode 100644
index 00000000..260ee66d
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/characteristics/uniformity.md
@@ -0,0 +1,63 @@
+# Uniformity
+
+The Uniformity
+
+## Mathematical Definition
+
+The uniformity can be calculated
+
+### from Congeneric Intervals Chains
+
+Let $CIC$ is [_Congenerics Intervals Chains_](../intervals_chains.md#as-matrix-ml) defined as matrix
+
+$$
+CIC =
+\begin{pmatrix}
+\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
+\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
+\end{pmatrix}
+$$
+
+$$u = \frac {1} {n} * \sum_{j=1}^{m}{\log_2 \frac{ \left(\sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}\right)^{n_j} } { \prod_{j=1}^{l} \Bigg\{\begin{array}{l}
+    \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
+    1, &   \Delta_{i,j} \in \{ - \}
+\end{array}} }$$
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
+    1, & \Delta_{i,j} \notin \{-\} \\
+    0, &   \Delta_{i,j} \in \{ - \}
+\end{array}}$ is count of _non-empty_ elements in $j$ _congeneric intervals chain_,
+$\Delta_{i,j}$ the $i$-th element of $j$-th _congeneric intervals chain_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
+
+
+### from Congenerics Intervals Distributions
+
+Let $CID$ is [_Congenerics Intervals Distributions_](../intervals_distribution.md#as-matrix-ml) defined as matrix
+
+$$
+CID =
+\begin{pmatrix}
+cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\
+cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l}
+\end{pmatrix}
+$$
+
+$$u = \frac {1} {n} * \sum_{j=1}^{m}{\log_2 \Bigg(\frac{ \sum_{i=1}^{l}{(i \times cid_{i,j})}^{n_j} } { \prod_{j=1}^{l} \Bigg\{\begin{array}{l}
+    i \times cid_{i,j}, & cid_{i,j} \neq 0 \\
+    1, &   cid_{i,j} = 0
+\end{array}} }\Bigg)$$
+
+
+where $m$ the _power_, $n_j = \sum_{i=1}^{l}{cid_{i,j}}$ is count of _non-empty_ elements in $j$ _congeneric intervals distribution_,
+$cid_{i,j}$ the $i$-th element of $j$-th _congeneric intervals distribution_.
+
+$$n=\sum_{j=1}^{m}{n_j}$$
diff --git a/docs/fundamentals/congeneric_decomposition/congeneric_intervals_distribution.md b/docs/fundamentals/congeneric_decomposition/congeneric_intervals_distribution.md
deleted file mode 100644
index a887f538..00000000
--- a/docs/fundamentals/congeneric_decomposition/congeneric_intervals_distribution.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Congeneric Intervals Distribution
-
-Coming soon
diff --git a/docs/fundamentals/congeneric_decomposition/congeneric_intervals_order.md b/docs/fundamentals/congeneric_decomposition/congeneric_intervals_order.md
deleted file mode 100644
index b86f13ae..00000000
--- a/docs/fundamentals/congeneric_decomposition/congeneric_intervals_order.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Congeneric Intervals Order
-
-Coming soon
diff --git a/docs/fundamentals/congeneric_decomposition/congeneric_intervals_tuple.md b/docs/fundamentals/congeneric_decomposition/congeneric_intervals_tuple.md
deleted file mode 100644
index 2423c53a..00000000
--- a/docs/fundamentals/congeneric_decomposition/congeneric_intervals_tuple.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Congeneric Intervals Tuple
-
-Coming soon
diff --git a/docs/fundamentals/congeneric_decomposition/congeneric_orders.md b/docs/fundamentals/congeneric_decomposition/congeneric_orders.md
deleted file mode 100644
index 57c3cd38..00000000
--- a/docs/fundamentals/congeneric_decomposition/congeneric_orders.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Congeneric Orders
-
-Coming soon
diff --git a/docs/fundamentals/congeneric_decomposition/congeneric_sequences.md b/docs/fundamentals/congeneric_decomposition/congeneric_sequences.md
deleted file mode 100644
index dfbdda11..00000000
--- a/docs/fundamentals/congeneric_decomposition/congeneric_sequences.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Congeneric Sequences
-
-Coming soon
diff --git a/docs/fundamentals/congeneric_decomposition/index.md b/docs/fundamentals/congeneric_decomposition/index.md
index 878292d3..912a6df4 100644
--- a/docs/fundamentals/congeneric_decomposition/index.md
+++ b/docs/fundamentals/congeneric_decomposition/index.md
@@ -1,3 +1,244 @@
 # Congeneric Decomposition
 
-Coming soon
+Any [_Sequence_](../order/sequence.md) (including [_Partial Sequence_](../partials_and_congenerics/sequence/index.md)) can be decomposed into $m$ (power of an [_Alphabet_](../order/alphabet.md)) [_Congeneric Sequences_](../partials_and_congenerics/sequence/congeneric.md) length of $l$. Each _congeneric sequence_ would be [_compatible_](../partials_and_congenerics/sequence/index.md#compatibility) with all other _congeneric sequences_, so the procedure is invertible. Result of the decomposition can be represented as a matrix $m \times l$
+
+The sequence
+
+``` mermaid
+block-beta
+  columns 36
+  seq1["I"] seq2["N"] seq3["T"] seq4["E"] seq5["L"] seq6["L"] seq7["I"] seq8["G"] seq9["E"] seq10["N"]
+  seq11["C"] seq12["E"] seq13[" "] seq14["I"] seq15["S"] seq16[" "] seq17["T"] seq18["H"] seq19["E"] seq20[" "]
+  seq21["A"] seq22["B"] seq23["I"] seq24["L"] seq25["I"] seq26["T"] seq27["Y"] seq28[" "] seq29["T"] seq30["O"]
+  seq31[" "] seq32["A"] seq33["D"] seq34["A"] seq35["P"] seq36["T"]
+
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+  class seq1,seq7,seq14,seq23,seq25,i1,i7,i14,i23,i25 c1
+  class seq2,seq10,n2,n10 c2
+  class seq3,seq17,seq26,seq29,seq36,t3,t17,t26,t29,t36 c3
+  class seq4,seq9,seq12,seq19,e4,e9,e12,e19  c4
+  class seq5,seq6,seq24,l5,l6,l24 c5
+  class seq8,g8 c6
+  class seq11,c11 c7
+  class seq13,seq16,seq20,seq28,seq31,sp13,sp16,sp20,sp28,sp31 c8
+  class seq15,s15 c9
+  class seq18,h18 c10
+  class seq21,seq32,seq34,a21,a32,a34 c11
+  class seq22,b22 c12
+  class seq27,y27 c13
+  class seq30,o30 c14
+  class seq33,d33 c15
+  class seq35,p35 c16
+```
+
+decomposed into
+
+``` mermaid
+block-beta
+  columns 37
+  diag["⟍"] col1["1"] col2["2"] space:16 col19["..."] space:16 col36["l"]
+  row1["1"] i1["I"] i2["-"] i3["-"] i4["-"] i5["-"] i6["-"] i7["I"] i8["-"] i9["-"] i10["-"]
+  i11["-"] i12["-"] i13["-"] i14["I"] i15["-"] i16["-"] i17["-"] i18["-"] i19["-"] i20["-"]
+  i21["-"] i22["-"] i23["I"] i24["-"] i25["I"] i26["-"] i27["-"] i28["-"] i29["-"] i30["-"]
+  i31["-"] i32["-"] i33["-"] i34["-"] i35["-"] i36["-"]
+  row2["2"] n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+  n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+  n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"] n30["-"]
+  n31["-"] n32["-"] n33["-"] n34["-"] n35["-"] n36["-"]
+  space t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+  t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+  t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"] t30["-"]
+  t31["-"] t32["-"] t33["-"] t34["-"] t35["-"] t36["T"]
+  space e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+  e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"]
+  e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"]
+  e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"]
+  space l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+  l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+  l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"] l30["-"]
+  l31["-"] l32["-"] l33["-"] l34["-"] l35["-"] l36["-"]
+  space g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+  g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+  g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"] g30["-"]
+  g31["-"] g32["-"] g33["-"] g34["-"] g35["-"] g36["-"]
+  space c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+  c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+  c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"] c30["-"]
+  c31["-"] c32["-"] c33["-"] c34["-"] c35["-"] c36["-"]
+  space sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+  sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+  sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28[" "] sp29["-"] sp30["-"]
+  sp31[" "] sp32["-"] sp33["-"] sp34["-"] sp35["-"] sp36["-"]
+  row9["⋮"] s1["-"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+  s11["-"] s12["-"] s13["-"] s14["-"] s15["S"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+  s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"] s30["-"]
+  s31["-"] s32["-"] s33["-"] s34["-"] s35["-"] s36["-"]
+  space h1["-"] h2["-"] h3["-"] h4["-"] h5["-"] h6["-"] h7["-"] h8["-"] h9["-"] h10["-"]
+  h11["-"] h12["-"] h13["-"] h14["-"] h15["-"] h16["-"] h17["-"] h18["H"] h19["-"] h20["-"]
+  h21["-"] h22["-"] h23["-"] h24["-"] h25["-"] h26["-"] h27["-"] h28["-"] h29["-"] h30["-"]
+  h31["-"] h32["-"] h33["-"] h34["-"] h35["-"] h36["-"]
+  space a1["-"] a2["-"] a3["-"] a4["-"] a5["-"] a6["-"] a7["-"] a8["-"] a9["-"] a10["-"]
+  a11["-"] a12["-"] a13["-"] a14["-"] a15["-"] a16["-"] a17["-"] a18["-"] a19["-"] a20["-"]
+  a21["A"] a22["-"] a23["-"] a24["-"] a25["-"] a26["-"] a27["-"] a28["-"] a29["-"] a30["-"]
+  a31["-"] a32["A"] a33["-"] a34["A"] a35["-"] a36["-"]
+  space b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+  b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+  b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"] b30["-"]
+  b31["-"] b32["-"] b33["-"] b34["-"] b35["-"] b36["-"]
+  space y1["-"] y2["-"] y3["-"] y4["-"] y5["-"] y6["-"] y7["-"] y8["-"] y9["-"] y10["-"]
+  y11["-"] y12["-"] y13["-"] y14["-"] y15["-"] y16["-"] y17["-"] y18["-"] y19["-"] y20["-"]
+  y21["-"] y22["-"] y23["-"] y24["-"] y25["-"] y26["-"] y27["Y"] y28["-"] y29["-"] y30["-"]
+  y31["-"] y32["-"] y33["-"] y34["-"] y35["-"] y36["-"]
+  space o1["-"] o2["-"] o3["-"] o4["-"] o5["-"] o6["-"] o7["-"] o8["-"] o9["-"] o10["-"]
+  o11["-"] o12["-"] o13["-"] o14["-"] o15["-"] o16["-"] o17["-"] o18["-"] o19["-"] o20["-"]
+  o21["-"] o22["-"] o23["-"] o24["-"] o25["-"] o26["-"] o27["-"] o28["-"] o29["-"] o30["O"]
+  o31["-"] o32["-"] o33["-"] o34["-"] o35["-"] o36["-"]
+  space d1["-"] d2["-"] d3["-"] d4["-"] d5["-"] d6["-"] d7["-"] d8["-"] d9["-"] d10["-"]
+  d11["-"] d12["-"] d13["-"] d14["-"] d15["-"] d16["-"] d17["-"] d18["-"] d19["-"] d20["-"]
+  d21["-"] d22["-"] d23["-"] d24["-"] d25["-"] d26["-"] d27["-"] d28["-"] d29["-"] d30["-"]
+  d31["-"] d32["-"] d33["D"] d34["-"] d35["-"] d36["-"]
+  row16["m"] p1["-"] p2["-"] p3["-"] p4["-"] p5["-"] p6["-"] p7["-"] p8["-"] p9["-"] p10["-"]
+  p11["-"] p12["-"] p13["-"] p14["-"] p15["-"] p16["-"] p17["-"] p18["-"] p19["-"] p20["-"]
+  p21["-"] p22["-"] p23["-"] p24["-"] p25["-"] p26["-"] p27["-"] p28["-"] p29["-"] p30["-"]
+  p31["-"] p32["-"] p33["-"] p34["-"] p35["P"] p36["-"]
+
+
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+  classDef text fill:#fff,color:#000,stroke-width:0px
+
+  class row1,row2,row9,row16 text
+  class col1,col2,col19,col36 text
+  class diag text
+
+
+  class i2,i3,i4,i5,i6,i8,i9,i10 text
+  class i11,i12,i13,i15,i16,i17,i18,i19,i20 text
+  class i21,i22,i24,i26,i27,i28,i29,i30 text
+  class i31,i32,i33,i34,i35,i36 text
+
+  class n1,n3,n4,n5,n6,n7,n8,n9 text
+  class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 text
+  class n21,n22,n23,n24,n25,n26,n27,n28,n29,n30 text
+  class n31,n32,n33,n34,n35,n36 text
+
+  class t1,t2,t4,t5,t6,t7,t8,t9,t10 text
+  class t11,t12,t13,t14,t15,t16,t18,t19,t20 text
+  class t21,t22,t23,t24,t25,t27,t28,t30 text
+  class t31,t32,t33,t34,t35 text
+
+  class e1,e2,e3,e5,e6,e7,e8,e10 text
+  class e11,e13,e14,e15,e16,e17,e18,e20 text
+  class e21,e22,e23,e24,e25,e26,e27,e28,e29,e30 text
+  class e31,e32,e33,e34,e35,e36 text
+
+  class l1,l2,l3,l4,l7,l8,l9,l10 text
+  class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 text
+  class l21,l22,l23,l25,l26,l27,l28,l29,l30 text
+  class l31,l32,l33,l34,l35,l36 text
+
+  class g1,g2,g3,g4,g5,g6,g7,g9,g10 text
+  class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 text
+  class g21,g22,g23,g24,g25,g26,g27,g28,g29,g30 text
+  class g31,g32,g33,g34,g35,g36 text
+
+  class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 text
+  class c12,c13,c14,c15,c16,c17,c18,c19,c20 text
+  class c21,c22,c23,c24,c25,c26,c27,c28,c29,c30 text
+  class c31,c32,c33,c34,c35,c36 text
+
+  class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 text
+  class sp11,sp12,sp14,sp15,sp17,sp18,sp19 text
+  class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29,sp30 text
+  class sp31,sp32,sp33,sp34,sp35,sp36 text
+
+  class s1,s2,s3,s4,s5,s6,s7,s8,s9,s10 text
+  class s11,s12,s13,s14,s16,s17,s18,s19,s20 text
+  class s21,s22,s23,s24,s25,s26,s27,s28,s29,s30 text
+  class s31,s32,s33,s34,s35,s36 text
+
+  class h1,h2,h3,h4,h5,h6,h7,h8,h9,h10 text
+  class h11,h12,h13,h14,h15,h16,h17,h19,h20 text
+  class h21,h22,h23,h24,h25,h26,h27,h28,h29,h30 text
+  class h31,h32,h33,h34,h35,h36 text
+
+
+  class a1,a2,a3,a4,a5,a6,a7,a8,a9,a10 text
+  class a11,a12,a13,a14,a15,a16,a17,a18,a19,a20 text
+  class a22,a23,a24,a25,a26,a27,a28,a29,a30 text
+  class a31,a33,a35,a36 text
+
+  class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 text
+  class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 text
+  class b21,b23,b24,b25,b26,b27,b28,b29,b30 text
+  class b31,b32,b33,b34,b35,b36 text
+
+  class y1,y2,y3,y4,y5,y6,y7,y8,y9,y10 text
+  class y11,y12,y13,y14,y15,y16,y17,y18,y19,y20 text
+  class y21,y22,y23,y24,y25,y26,y28,y29,y30 text
+  class y31,y32,y33,y34,y35,y36 text
+
+  class o1,o2,o3,o4,o5,o6,o7,o8,o9,o10 text
+  class o11,o12,o13,o14,o15,o16,o17,o18,o19,o20 text
+  class o21,o22,o23,o24,o25,o26,o27,o28,o29 text
+  class o31,o32,o33,o34,o35,o36 text
+
+  class d1,d2,d3,d4,d5,d6,d7,d8,d9,d10 text
+  class d11,d12,d13,d14,d15,d16,d17,d18,d19,d20 text
+  class d21,d22,d23,d24,d25,d26,d27,d28,d29,d30 text
+  class d31,d32,d34,d35,d36 text
+
+  class p1,p2,p3,p4,p5,p6,p7,p8,p9,p10 text
+  class p11,p12,p13,p14,p15,p16,p17,p18,p19,p20 text
+  class p21,p22,p23,p24,p25,p26,p27,p28,p29,p30 text
+  class p31,p32,p33,p34,p36 text
+
+  class seq1,seq7,seq14,seq23,seq25,i1,i7,i14,i23,i25 c1
+  class seq2,seq10,n2,n10 c2
+  class seq3,seq17,seq26,seq29,seq36,t3,t17,t26,t29,t36 c3
+  class seq4,seq9,seq12,seq19,e4,e9,e12,e19  c4
+  class seq5,seq6,seq24,l5,l6,l24 c5
+  class seq8,g8 c6
+  class seq11,c11 c7
+  class seq13,seq16,seq20,seq28,seq31,sp13,sp16,sp20,sp28,sp31 c8
+  class seq15,s15 c9
+  class seq18,h18 c10
+  class seq21,seq32,seq34,a21,a32,a34 c11
+  class seq22,b22 c12
+  class seq27,y27 c13
+  class seq30,o30 c14
+  class seq33,d33 c15
+  class seq35,p35 c16
+```
+
+Applying [operations](../order/index.md) to [_Congeneric sequences_](./sequences.md) and its derivatives produces [_Congenerics Orders_](./orders.md), [_Congenerics Intervals Chains_](./intervals_chains.md),
+[_Congenerics Intervals Distributions_](./intervals_distribution.md) and finally [_Characteristics_](./characteristics/index.md) specific for _congeneric_ matrix.
diff --git a/docs/fundamentals/congeneric_decomposition/intervals_chains.md b/docs/fundamentals/congeneric_decomposition/intervals_chains.md
new file mode 100644
index 00000000..5c21222c
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/intervals_chains.md
@@ -0,0 +1,245 @@
+# Congeneric Intervals Chains
+
+_Congeneric Intervals Chains_ is a stack of a [_compatible_](../partials_and_congenerics/sequence/index.md#compatibility) [_congeric interval chain_](../partials_and_congenerics/intervals_chain/congeneric.md)
+that represents [_Intervals Chain_](../order/intervals_chain/index.md) or [_Partial Intervals Chain_](../partials_and_congenerics/intervals_chain/index.md). The _congeneric intervals chains_ are sorted in the stack by the first position with a _non-empty_ element in it.
+
+``` mermaid
+block-beta
+columns 30
+
+space i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+j1["1"] s1["23"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["6"] s8["-"] s9["-"] s10["-"]
+s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"]
+
+j2["2"] n1["-"] n2["21"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["8"]
+n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"]
+
+j3["3"] t1["-"] t2["-"] t3["3"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["14"] t18["-"] t19["-"] t20["-"]
+t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["9"] t27["-"] t28["-"] t29["3"]
+
+j4["4"] e1["-"] e2["-"] e3["-"] e4["21"] e5["-"] e6["-"] e7["-"] e8["-"] e9["5"] e10["-"]
+e11["-"] e12["3"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["-"] e20["-"]
+e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"]
+
+j5["5"] l1["-"] l2["-"] l3["-"] l4["-"] l5["10"] l6["1"] l7["-"] l8["-"] l9["-"] l10["-"]
+l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+l21["-"] l22["-"] l23["-"] l24["18"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"]
+
+j6["6"] g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["29"] g9["-"] g10["-"]
+g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"]
+
+j7["7"] c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+c11["29"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"]
+
+j8["8"] sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+sp11["-"] sp12["-"] sp13["22"] sp14["-"] sp15["-"] sp16["3"] sp17["-"] sp18["-"] sp19["-"] sp20["4"]
+sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28["-"] sp29["-"]
+
+j9["9"] b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+b21["-"] b22["29"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"]
+
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s2,s3,s4,s5,s6,s8,s9,s10 index
+class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 index
+class s21,s22,s23,s24,s25,s26,s27,s28,s29 index
+
+class n1,n3,n4,n5,n6,n7,n8,n9 index
+class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 index
+class n21,n22,n23,n24,n25,n26,n27,n28,n29 index
+
+class t1,t2,t4,t5,t6,t7,t8,t9,t10 index
+class t11,t12,t13,t14,t15,t16,t18,t19,t20 index
+class t21,t22,t23,t24,t25,t27,t28 index
+
+class e1,e2,e3,e5,e6,e7,e8,e10 index
+class e11,e13,e14,e15,e16,e17,e18,e19,e20 index
+class e21,e22,e23,e24,e25,e26,e27,e28,e29 index
+
+class l1,l2,l3,l4,l7,l8,l9,l10 index
+class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 index
+class l21,l22,l23,l25,l26,l27,l28,l29 index
+
+class g1,g2,g3,g4,g5,g6,g7,g9,g10 index
+class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 index
+class g21,g22,g23,g24,g25,g26,g27,g28,g29 index
+
+class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 index
+class c12,c13,c14,c15,c16,c17,c18,c19,c20 index
+class c21,c22,c23,c24,c25,c26,c27,c28,c29 index
+
+class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 index
+class sp11,sp12,sp14,sp15,sp17,sp18,sp19 index
+class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29 index
+
+class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 index
+class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 index
+class b21,b23,b24,b25,b26,b27,b28,b29 index
+
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+
+class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+```
+
+## Mathematical Definition
+
+Let $IC_{c}$ is [_Congeneric Interval Chain_](../partials_and_congenerics/intervals_chain/congeneric.md#mathematical-definition)
+
+Define a _Congeric intervals chains_
+
+### as m-tuple of congeneric intervals chain
+
+Let $compatible(IC1_c, IC2_c)$ is [_compatibility of Partial Sequences function_](../partials_and_congenerics/sequence/index.md#compatibility)
+
+$CIC$ is m-tuple of $IC_c$ _intervals chain_
+
+$$CIC = <cic_1, cic_2, ..., cic_m>,$$
+
+$$\forall j \in \{1, ..., m\} \exists cic_j \in \{IC_c\}, $$
+
+all _congeneric intervals chains_ are _compatible_
+
+$$compatible(CIC(k), CIC(j)) \bigg| \forall k,j \in \{1, ..., m\}, k \ne j,$$
+
+and sorted by first apperance of _non-empty_ element
+
+$$\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, CIC(k)(x) \ne 1 \bigg| CIC(j)(i) \notin \{-\} \land CIC(j)(x) \in \{-\}$$
+
+where:
+
+- $cic_j$​ is called the $j$-th _congeneric intervals chain_.
+- $l := |CIC(1)|$ is _length_, $l \in N$
+- $m := |CIC|$ is _power_, $m \in N$
+
+### as matrix (m,l)
+
+$CIC$ is (m,l) matrix of $\{1,..,, l\} \cup \{-\}$
+
+$$
+CIC =
+\begin{pmatrix}
+\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
+\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
+\end{pmatrix}
+$$
+
+$$\forall j \in \{1, ..., m\}, i \in \{1, ..., l\} \exists \Delta_{j,i} \in \{1,..,, l\} \cup \{-\}, $$
+
+columns are _compatible_
+
+$$\ \Delta_{k,i} \in \{-\} \big | \forall i \in \{1,...,l\}, \forall j \in \{1,...,m\}, \forall  k \ne j,\ \Delta_{j,i} \in \{1,...,l\} $$
+
+and sorted by first apperance of _non-empty_ element
+
+$$\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, \Delta_{k,x} \in \{-\} \bigg| \Delta_{j,i} \in \{1,...,l\} \land \Delta_{j,x} \in \{-\}$$
+
+where:
+
+- $\Delta_j$​ is called the $j$-th _congeneric intervals chain_ or $j$-th _row_.
+- $\Delta_{j,i}$​ is called the $i$-th element of $j$-th _congeneric intervals chain_.
+- $l$ is _length_, $l \in N$
+- $m$ is _power_, $m \in N$
+
+
+### from Congeneric Sequences
+
+Let $X_{-}$ is a [_Parial carrier set_](../partials_and_congenerics/carrier_set.md#mathematical-definition)
+
+Let $CS$ is [_Congeneric Sequences_](./sequences.md#mathematical-definition) - (m,l) matrix of $X_{-}$
+
+Let $Binging_p$ is [_Partial Binding_](../partials_and_congenerics/intervals_chain/index.md#define-partial-bindings)
+
+Let $Intervals_p$ is [_Partial Intervals function_](../partials_and_congenerics/intervals_chain/index.md#define-partial-intervals-chain) described as function
+
+$$Intervals_p : \big\{Binding_p\big\} \times \big\{S_p\}  \longrightarrow \big\{ IC_p \big\}$$
+
+$$
+CS =
+\begin{pmatrix}
+cs_{1,1} & cs_{1,2} & \cdots & cs_{1,l} \\
+cs_{2,1} & cs_{2,2} & \cdots & cs_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cs_{m,1} & cs_{m,2} & \cdots & cs_{m,l}
+\end{pmatrix}
+$$
+
+Define
+
+$$congenerics\_intervals\_chains : \{CS\} \longrightarrow \{CIC\}$$
+
+$$congenerics\_intervals\_chains(CS)(j)(i) = Intervals_p(CS(j))(i) $$
+
+$$\exists congenerics\_intervals\_chain^{-1} : \{CIC\} \longrightarrow \{CS\},$$
+
+$$congenerics\_intervals\_chains^{-1}(CIC)(i) = \Bigg\{\begin{array}{l}
+    j, & \exists j \in \{1,...,m\}, CIC(j)(i) \notin \{-\} \\
+    -, &   otherwise
+\end{array}$$
+
+
+### from Congeneric Orders
+
+Let $CO$ is [_Congeneric Orders_](./orders.md#mathematical-definition) - (m,l) matrix of $\{1,...,l\} \cup \{-\}$
+
+Let $Binging_p$ is [_Partial Binding_](../partials_and_congenerics/intervals_chain/index.md#define-partial-bindings)
+
+Let $Intervals_p$ is [_Partial Intervals function_](../partials_and_congenerics/intervals_chain/index.md#define-partial-intervals-chain) described as function
+
+$$Intervals_p : \big\{Binding_p\big\} \times \big\{S_p\}  \longrightarrow \big\{ IC_p \big\}$$
+
+$$
+CO =
+\begin{pmatrix}
+co_{1,1} & co_{1,2} & \cdots & co_{1,l} \\
+co_{2,1} & co_{2,2} & \cdots & co_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+co_{m,1} & co_{m,2} & \cdots & co_{m,l}
+\end{pmatrix}
+$$
+
+Define
+
+$$congenerics\_intervals\_chains : \{CO\} \longrightarrow \{CIC\}$$
+
+$$congenerics\_intervals\_chains(CO)(j)(i) = Intervals_p(CO(j))(i) $$
+
+$$\exists congenerics\_intervals\_chain^{-1} : \{CIC\} \longrightarrow \{CO\},$$
+
+$$congenerics\_intervals\_chains^{-1}(CIC)(i) = \Bigg\{\begin{array}{l}
+    1, & \exists j \in \{1,...,m\}, CIC(j)(i) \notin \{-\} \\
+    -, &   otherwise
+\end{array}$$
diff --git a/docs/fundamentals/congeneric_decomposition/intervals_distribution.md b/docs/fundamentals/congeneric_decomposition/intervals_distribution.md
new file mode 100644
index 00000000..f188b391
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/intervals_distribution.md
@@ -0,0 +1,184 @@
+# Congeneric Intervals Distributions
+
+_Congeneric Intervals Distributions_ is a stack of [_Partials Intervals Distribution_](../partials_and_congenerics/intervals_distribution.md)
+that represents [_Intervals Distribution_](../order/intervals_distribution/index.md).
+
+``` mermaid
+block-beta
+columns 30
+
+space i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+j1["1"] s1["0"] s2["0"] s3["0"] s4["0"] s5["0"] s6["1"] s7["0"] s8["0"] s9["0"] s10["0"]
+s11["0"] s12["0"] s13["0"] s14["0"] s15["0"] s16["0"] s17["0"] s18["0"] s19["0"] s20["0"]
+s21["0"] s22["0"] s23["1"] s24["0"] s25["0"] s26["0"] s27["0"] s28["0"] s29["0"]
+
+j2["2"] n1["0"] n2["0"] n3["0"] n4["0"] n5["0"] n6["0"] n7["0"] n8["1"] n9["0"] n10["0"]
+n11["0"] n12["0"] n13["0"] n14["0"] n15["0"] n16["0"] n17["0"] n18["0"] n19["0"] n20["0"]
+n21["1"] n22["0"] n23["0"] n24["0"] n25["0"] n26["0"] n27["0"] n28["0"] n29["0"]
+
+j3["3"] t1["0"] t2["0"] t3["2"] t4["0"] t5["0"] t6["0"] t7["0"] t8["0"] t9["1"] t10["0"]
+t11["0"] t12["0"] t13["0"] t14["1"] t15["0"] t16["0"] t17["0"] t18["0"] t19["0"] t20["0"]
+t21["0"] t22["0"] t23["0"] t24["0"] t25["0"] t26["0"] t27["0"] t28["0"] t29["0"]
+
+j4["4"] e1["0"] e2["0"] e3["1"] e4["0"] e5["1"] e6["0"] e7["0"] e8["0"] e9["0"] e10["0"]
+e11["0"] e12["0"] e13["0"] e14["0"] e15["0"] e16["0"] e17["0"] e18["0"] e19["0"] e20["0"]
+e21["1"] e22["0"] e23["0"] e24["0"] e25["0"] e26["0"] e27["0"] e28["0"] e29["0"]
+
+j5["5"] l1["1"] l2["0"] l3["0"] l4["0"] l5["0"] l6["0"] l7["0"] l8["0"] l9["0"] l10["1"]
+l11["0"] l12["0"] l13["0"] l14["0"] l15["0"] l16["0"] l17["0"] l18["1"] l19["0"] l20["0"]
+l21["0"] l22["0"] l23["0"] l24["0"] l25["0"] l26["0"] l27["0"] l28["0"] l29["0"]
+
+j6["6"] g1["0"] g2["0"] g3["0"] g4["0"] g5["0"] g6["0"] g7["0"] g8["0"] g9["0"] g10["0"]
+g11["0"] g12["0"] g13["0"] g14["0"] g15["0"] g16["0"] g17["0"] g18["0"] g19["0"] g20["0"]
+g21["0"] g22["0"] g23["0"] g24["0"] g25["0"] g26["0"] g27["0"] g28["0"] g29["1"]
+
+j7["7"] c1["0"] c2["0"] c3["0"] c4["0"] c5["0"] c6["0"] c7["0"] c8["0"] c9["0"] c10["0"]
+c11["0"] c12["0"] c13["0"] c14["0"] c15["0"] c16["0"] c17["0"] c18["0"] c19["0"] c20["0"]
+c21["0"] c22["0"] c23["0"] c24["0"] c25["0"] c26["0"] c27["0"] c28["0"] c29["1"]
+
+j8["8"] sp1["0"] sp2["0"] sp3["1"] sp4["1"] sp5["0"] sp6["0"] sp7["0"] sp8["0"] sp9["0"] sp10["0"]
+sp11["0"] sp12["0"] sp13["0"] sp14["0"] sp15["0"] sp16["0"] sp17["0"] sp18["0"] sp19["0"] sp20["0"]
+sp21["0"] sp22["1"] sp23["0"] sp24["0"] sp25["0"] sp26["0"] sp27["0"] sp28["0"] sp29["0"]
+
+j9["9"] b1["0"] b2["0"] b3["0"] b4["0"] b5["0"] b6["0"] b7["0"] b8["0"] b9["0"] b10["0"]
+b11["0"] b12["0"] b13["0"] b14["0"] b15["0"] b16["0"] b17["0"] b18["0"] b19["0"] b20["0"]
+b21["0"] b22["0"] b23["0"] b24["0"] b25["0"] b26["0"] b27["0"] b28["0"] b29["1"]
+
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s1,s2,s3,s4,s5,s7,s8,s9,s10 index
+class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 index
+class s21,s22,s24,s25,s26,s27,s28,s29 index
+
+class n1,n2,n3,n4,n5,n6,n7,n9,n10 index
+class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 index
+class n22,n23,n24,n25,n26,n27,n28,n29 index
+
+class t1,t2,t4,t5,t6,t7,t8,t10 index
+class t11,t12,t13,t15,t16,t17,t18,t19,t20 index
+class t21,t22,t23,t24,t25,t26,t27,t28,t29 index
+
+class e1,e2,e4,e6,e7,e8,e9,e10 index
+class e11,e12,e13,e14,e15,e16,e17,e18,e19,e20 index
+class e22,e23,e24,e25,e26,e27,e28,e29 index
+
+class l2,l3,l4,l5,l6,l7,l8,l9 index
+class l11,l12,l13,l14,l15,l16,l17,l19,l20 index
+class l21,l22,l23,l24,l25,l26,l27,l28,l29 index
+
+class g1,g2,g3,g4,g5,g6,g7,g8,g9,g10 index
+class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 index
+class g21,g22,g23,g24,g25,g26,g27,g28 index
+
+class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 index
+class c11,c12,c13,c14,c15,c16,c17,c18,c19,c20 index
+class c21,c22,c23,c24,c25,c26,c27,c28 index
+
+class sp1,sp2,sp5,sp6,sp7,sp8,sp9,sp10 index
+class sp11,sp12,sp13,sp14,sp15,sp16,sp17,sp18,sp19,sp20 index
+class sp21,sp23,sp24,sp25,sp26,sp27,sp28,sp29 index
+
+class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 index
+class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 index
+class b21,b22,b23,b24,b25,b26,b27,b28 index
+
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+
+class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+```
+
+## Mathematical Definition
+
+Let $ID_p$ is [_Partial Interval Distribution_](../partials_and_congenerics/intervals_distribution.md#mathematical-definition)
+
+Define a _Congeric intervals distributions_
+
+### as m-tuple of congeneric intervals distributions
+
+$CID$ is m-tuple of $ID_c$ _intervals distributions_
+
+$$CID = <cid_1, cid_2, ..., cid_m>,$$
+
+$$\forall j \in \{1, ..., m\} \exists cid_j \in \{ID_c\}, $$
+
+where:
+
+- $cid_j$​ is called the $j$-th _congeneric intervals distribution_.
+- $l := |CID(1)|$ is _length_, $l \in N$
+- $m := |CID|$ is _power_, $m \in N$
+
+### as matrix (m,l)
+
+$CID$ is (m,l) matrix of $\{0,..,, l\}$
+
+$$
+CID =
+\begin{pmatrix}
+cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\
+cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l}
+\end{pmatrix}
+$$
+
+$$\forall j \in \{1, ..., m\}, i \in \{1, ..., l\} \exists cid_{j,i} \in \{0,..,, l\}, $$
+
+where:
+
+- $cid_j$​ is called the $j$-th _congeneric intervals distribution_ or $j$-th _row_.
+- $cid_{j,i}$​ is called the $i$-th element of $j$-th _congeneric intervals distribution_.
+- $l$ is _length_, $l \in N$
+- $m$ is _power_, $m \in N$
+
+
+### from Congeneric Intervals Chains
+
+Let $IC_{p}$ is [_Partial Interval Chain_](../partials_and_congenerics/intervals_chain/index.md#define-partial-intervals-chain)
+
+Let $ID_p$ is [_Partial Intervals distribution_](../partials_and_congenerics/intervals_distribution.md#mathematical-definition) described as function
+
+$$ID_p : \big\{  IC_p \big\} \longrightarrow \big\{ \{1,...,l\} \longrightarrow  N_0 \big\},$$
+
+Let $CIC$ is [_Congeneric Intervals Chains_](./intervals_chains.md#as-matrix-ml) - (m,l) matrix of $\{1,...,l\} \cup \{-\}$
+
+$$
+CIC =
+\begin{pmatrix}
+\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
+\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
+\end{pmatrix}
+$$
+
+Define
+
+$$congenerics\_intervals\_distributions : \{CIC\} \longrightarrow \{CID\}$$
+
+$$congenerics\_intervals\_distributions(CIC)(j)(i) = ID_p(CIC(j))(i) $$
diff --git a/docs/fundamentals/congeneric_decomposition/orders.md b/docs/fundamentals/congeneric_decomposition/orders.md
new file mode 100644
index 00000000..f0c980a6
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/orders.md
@@ -0,0 +1,262 @@
+# Congeneric Orders
+
+_Congeneric orders_ is a stack of a [_compatible_](../partials_and_congenerics/sequence/index.md#compatibility) [_congeric orders_](../partials_and_congenerics/order/congeneric.md)
+that represents [_Order_](../order/order.md) or [_Partial Order_](../partials_and_congenerics/order/index.md). The _congeneric orders_ are sorted in the stack by the first position with a _non-empty_ element in it.
+
+``` mermaid
+block-beta
+columns 30
+
+space i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+j1["1"] s1["1"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["1"] s8["-"] s9["-"] s10["-"]
+s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"]
+
+j2["2"] n1["-"] n2["1"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["1"]
+n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"]
+
+j3["3"] t1["-"] t2["-"] t3["1"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["1"] t18["-"] t19["-"] t20["-"]
+t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["1"] t27["-"] t28["-"] t29["1"]
+
+j4["4"] e1["-"] e2["-"] e3["-"] e4["1"] e5["-"] e6["-"] e7["-"] e8["-"] e9["1"] e10["-"]
+e11["-"] e12["1"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["-"] e20["-"]
+e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"]
+
+j5["5"] l1["-"] l2["-"] l3["-"] l4["-"] l5["1"] l6["1"] l7["-"] l8["-"] l9["-"] l10["-"]
+l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+l21["-"] l22["-"] l23["-"] l24["1"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"]
+
+j6["6"] g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["1"] g9["-"] g10["-"]
+g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"]
+
+j7["7"] c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+c11["1"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"]
+
+j8["8"] sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+sp11["-"] sp12["-"] sp13["1"] sp14["-"] sp15["-"] sp16["1"] sp17["-"] sp18["-"] sp19["-"] sp20["1"]
+sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28["-"] sp29["-"]
+
+j9["9"] b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+b21["-"] b22["1"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"]
+
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s2,s3,s4,s5,s6,s8,s9,s10 index
+class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 index
+class s21,s22,s23,s24,s25,s26,s27,s28,s29 index
+
+class n1,n3,n4,n5,n6,n7,n8,n9 index
+class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 index
+class n21,n22,n23,n24,n25,n26,n27,n28,n29 index
+
+class t1,t2,t4,t5,t6,t7,t8,t9,t10 index
+class t11,t12,t13,t14,t15,t16,t18,t19,t20 index
+class t21,t22,t23,t24,t25,t27,t28 index
+
+class e1,e2,e3,e5,e6,e7,e8,e10 index
+class e11,e13,e14,e15,e16,e17,e18,e19,e20 index
+class e21,e22,e23,e24,e25,e26,e27,e28,e29 index
+
+class l1,l2,l3,l4,l7,l8,l9,l10 index
+class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 index
+class l21,l22,l23,l25,l26,l27,l28,l29 index
+
+class g1,g2,g3,g4,g5,g6,g7,g9,g10 index
+class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 index
+class g21,g22,g23,g24,g25,g26,g27,g28,g29 index
+
+class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 index
+class c12,c13,c14,c15,c16,c17,c18,c19,c20 index
+class c21,c22,c23,c24,c25,c26,c27,c28,c29 index
+
+class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 index
+class sp11,sp12,sp14,sp15,sp17,sp18,sp19 index
+class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29 index
+
+class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 index
+class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 index
+class b21,b23,b24,b25,b26,b27,b28,b29 index
+
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+
+class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+```
+
+## Mathematical Definition
+
+Let $X_{-}$ is [_Partial Carrier set_](../partials_and_congenerics/carrier_set.md#mathematical-definition)
+
+Let $O_{c}$ is [_Congeneric Order_](../partials_and_congenerics/order/congeneric.md#mathematical-definition)
+
+Define a _Congeric orders_
+
+### as m-tuple of congeneric order
+
+Let $compatible(O1_c, O2_c)$ is [_compatibility of Partial Sequences function_](../partials_and_congenerics/sequence/index.md#compatibility)
+
+$CO$ is m-tuple of $O_c$ orders
+
+$$CO = <co_1, co_2, ..., co_m>,$$
+
+$$\forall j \in \{1, ..., m\} \exists co_j \in \{O_c\}, $$
+
+all _congeneric orders_ are _compatible_
+
+$$compatible(CO(k), CO(j)) \bigg| \forall k,j \in \{1, ..., m\}, k \ne j,$$
+
+and sorted by first apperance of _non-empty_ element
+
+$$\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, CO(k)(x) \ne 1 \bigg| CO(j)(i) = 1 \land CO(j)(x) \ne 1$$
+
+where:
+
+- $co_j$​ is called the $j$-th _congeneric order_.
+- $l := |CO(1)|$ is _length_, $l \in N$
+- $m := |CO|$ is _power_, $m \in N$
+
+### as matrix (m,l)
+
+$CO$ is (m,l) matrix of $\{-, 1\}$
+
+$$
+C0 =
+\begin{pmatrix}
+co_{1,1} & co_{1,2} & \cdots & co_{1,l} \\
+co_{2,1} & co_{2,2} & \cdots & co_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+co_{m,1} & co_{m,2} & \cdots & co_{m,l}
+\end{pmatrix}
+$$
+
+$$\forall j \in \{1, ..., m\}, i \in \{1, ..., l\} \exists co_{j,i} \in \{-, 1\}, $$
+
+columns are _compatible_
+
+$$\ co_{k,i} \ne 1 \big | \forall i \in \{1,...,l\}, \forall j \in \{1,...,m\}, \forall  k \ne j,\ co_{j,i} = 1 $$
+
+and sorted by first apperance of _non-empty_ element
+
+$$\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, co_{k,x} \ne 1 \bigg| co_{j,i} = 1 \land co_{j,x} \ne 1$$
+
+where:
+
+- $co_j$​ is called the $j$-th _congeneric order or $j$-th _row_.
+- $co_{j,i}$​ is called the $i$-th element of $j$-th _congeneric order_.
+- $l$ is _length_, $l \in N$
+- $m$ is _power_, $m \in N$
+
+### from Order
+
+Let $O_p$ is [_Partial Order_](../partials_and_congenerics/order/index.md#mathematical-definition) defined as function
+
+$$O_p : \{1,...,l\} \longrightarrow \{1,...,l\} \cup \{-\}$$
+
+$$alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+Define
+
+$$congenerics\_orders : \{O_p\} \longrightarrow \{CO\}$$
+
+$$
+congenerics\_orders(O_p)(j)(i) = \Bigg\{\begin{array}{l}
+    1 &   O_p(i)=j \\
+    -, &   O_p(i) \in \{-\} \lor O_p(i) \ne j
+\end{array}
+$$
+
+$$\exists congenerics\_orders^{-1} : \{CO\} \longrightarrow \{S_p\},$$
+
+$$
+congenerics\_orders^{-1}(CO)(i) = \Bigg\{\begin{array}{l}
+    j, & \exists j \in \{1,...,m\}, CS(j)(i) = 1 \\
+    -,      &   otherwise
+\end{array}
+$$
+
+### from Congeneric Sequences
+
+Let $CS$ is [_Congeneric Sequences_](./sequences.md#mathematical-definition) - (m,l) matrix of $X_{-}$
+
+$$
+CS =
+\begin{pmatrix}
+cs_{1,1} & cs_{1,2} & \cdots & cs_{1,l} \\
+cs_{2,1} & cs_{2,2} & \cdots & cs_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cs_{m,1} & cs_{m,2} & \cdots & cs_{m,l}
+\end{pmatrix}
+$$
+
+Define
+
+$$congenerics\_orders : \{CS\} \longrightarrow \{CO\}$$
+
+$$congenerics\_orders(CS)(j)(i) = \Bigg\{\begin{array}{l}
+    1, &   CS(i) \notin \{-\}  \\
+    -, &   CS(i) \in \{-\}
+\end{array}$$
+
+$$\exists congenerics\_orders^{-1} : \{CO\} \longrightarrow \{CS\},$$
+
+$$congenerics\_orders^{-1}(CO)(i) = \Bigg\{\begin{array}{l}
+    j, & \exists j \in \{1,...,m\}, CO(j)(i) = 1 \\
+    -, &   otherwise
+\end{array}$$
+
+### Congeneric Orders product Alphabet
+
+Let $X_{-}$ is a [_Parial carrier set_](../partials_and_congenerics/carrier_set.md#mathematical-definition)
+
+Let $A$ is a [_Aphabet_](./alphabet.md) $A : \{1, ..., m\} \longrightarrow X,$
+
+Let $alphabet_p$ is [_Alphabet of Partial Sequence function_](../partials_and_congenerics/alphabet/index.md#mathematical-definition)
+
+$$alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+Let $CS$ is a [_Congeneric Sequeneces_](./sequences.md#mathematical-definition)
+
+the following equations are true
+
+$$CO = congenerics\_orders(CS),$$
+
+$$A = alphabet_p(CS),$$
+
+$$CS = ( CO \odot A ),$$
+
+$$
+CS(j)(i) = \Bigg\{\begin{array}{l}
+    A(j), & \exists j \in \{1,...,m\}, CO(j)(i) = 1 \\
+    -, &   otherwise
+\end{array}
+$$
diff --git a/docs/fundamentals/congeneric_decomposition/sequences.md b/docs/fundamentals/congeneric_decomposition/sequences.md
new file mode 100644
index 00000000..fe9f3a4e
--- /dev/null
+++ b/docs/fundamentals/congeneric_decomposition/sequences.md
@@ -0,0 +1,199 @@
+# Congeneric Sequences
+
+_Congeneric sequences_ is a stack of a [_compatible_](../partials_and_congenerics/sequence/index.md#compatibility) [_congeric sequnces_](../partials_and_congenerics/sequence/congeneric.md)
+that represents [_Sequence_](../order/sequence.md) or [_Partial Sequence_](../partials_and_congenerics/sequence/index.md). The _congeneric sequnces_ are ordered in the stack by the first position with a _non-empty_ element in it.
+
+``` mermaid
+block-beta
+columns 30
+
+space i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+j1["1"] s1["I"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["I"] s8["-"] s9["-"] s10["-"]
+s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"]
+
+j2["2"] n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"]
+
+j3["3"] t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"]
+
+j4["4"] e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["-"] e20["-"]
+e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"]
+
+j5["5"] l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"]
+
+j6["6"] g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"]
+
+j7["7"] c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"]
+
+j8["8"] sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28["-"] sp29["-"]
+
+j9["9"] b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"]
+
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s2,s3,s4,s5,s6,s8,s9,s10 index
+class s11,s12,s13,s14,s15,s16,s17,s18,s19,s20 index
+class s21,s22,s23,s24,s25,s26,s27,s28,s29 index
+
+class n1,n3,n4,n5,n6,n7,n8,n9 index
+class n11,n12,n13,n14,n15,n16,n17,n18,n19,n20 index
+class n21,n22,n23,n24,n25,n26,n27,n28,n29 index
+
+class t1,t2,t4,t5,t6,t7,t8,t9,t10 index
+class t11,t12,t13,t14,t15,t16,t18,t19,t20 index
+class t21,t22,t23,t24,t25,t27,t28 index
+
+class e1,e2,e3,e5,e6,e7,e8,e10 index
+class e11,e13,e14,e15,e16,e17,e18,e19,e20 index
+class e21,e22,e23,e24,e25,e26,e27,e28,e29 index
+
+class l1,l2,l3,l4,l7,l8,l9,l10 index
+class l11,l12,l13,l14,l15,l16,l17,l18,l19,l20 index
+class l21,l22,l23,l25,l26,l27,l28,l29 index
+
+class g1,g2,g3,g4,g5,g6,g7,g9,g10 index
+class g11,g12,g13,g14,g15,g16,g17,g18,g19,g20 index
+class g21,g22,g23,g24,g25,g26,g27,g28,g29 index
+
+class c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 index
+class c12,c13,c14,c15,c16,c17,c18,c19,c20 index
+class c21,c22,c23,c24,c25,c26,c27,c28,c29 index
+
+class sp1,sp2,sp3,sp4,sp5,sp6,sp7,sp8,sp9,sp10 index
+class sp11,sp12,sp14,sp15,sp17,sp18,sp19 index
+class sp21,sp22,sp23,sp24,sp25,sp26,sp27,sp28,sp29 index
+
+class b1,b2,b3,b4,b5,b6,b7,b8,b9,b10 index
+class b11,b12,b13,b14,b15,b16,b17,b18,b19,b20 index
+class b21,b23,b24,b25,b26,b27,b28,b29 index
+
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+
+class j1,j2,j3,j4,j5,j6,j7,j8,j9 index
+```
+
+## Mathematical Definition
+
+Let $X_{-}$ is [_Partial Carrier set_](../partials_and_congenerics/carrier_set.md#mathematical-definition)
+
+Let $S_{c}$ is [_Congeneric Sequence_](../partials_and_congenerics/sequence/congeneric.md#mathematical-definition)
+
+Define a _Congeric sequences_
+
+### as m-tuple of congeneric sequences
+
+Let $compatible(S1_c, S2_c)$ is [_compatibility of Partial Sequences function_](../partials_and_congenerics/sequence/index.md#compatibility)
+
+$CS$ is m-tuple of $S_c$ sequences
+
+$$CS = <cs_1, cs_2, ..., cs_m>,$$
+
+$$\forall j \in \{1, ..., m\} \exists cs_j \in \{S_c\}, $$
+
+all _congeneric sequences_ are _compatible_
+
+$$compatible(CS(k), CS(j)) \bigg| \forall k,j \in \{1, ..., m\}, k \ne j,$$
+
+and sorted by first apperance of _non-empty_ element
+
+$$\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, CS(k)(x) \in \{-\} \bigg| CS(j)(i) \notin \{-\} \land CS(j)(x) \in \{-\}$$
+
+where:
+
+- $cs_j$​ is called the $j$-th _congeneric sequence_.
+- $l := |CS(1)|$ is _length_, $l \in N$
+- $m := |CS|$ is _power_, $m \in N$
+
+### as matrix (m,l)
+
+$CS$ is (m,l) matrix of $X_{-}$
+
+$$
+CS =
+\begin{pmatrix}
+cs_{1,1} & cs_{1,2} & \cdots & cs_{1,l} \\
+cs_{2,1} & cs_{2,2} & \cdots & cs_{2,l} \\
+\vdots   & \vdots   & \ddots & \vdots   \\
+cs_{m,1} & cs_{m,2} & \cdots & cs_{m,l}
+\end{pmatrix}
+$$
+
+$$\forall j \in \{1, ..., m\}, i \in \{1, ..., l\} \exists cs_{j,i} \in X_{-}, $$
+
+columns are _compatible_
+
+$$\ cs_{k,i} \in \{-\} \big | \forall i \in \{1,...,l\}, \forall j \in \{1,...,m\}, \forall  k \ne j,\ cs_{j,i} \notin \{-\} $$
+
+and sorted by first apperance of _non-empty_ element
+
+$$\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, cs_{k,x} \in \{-\} \bigg| cs_{j,i} \notin \{-\} \land cs_{j,x} \in \{-\}$$
+
+where:
+
+- $cs_j$​ is called the $j$-th _congeneric sequence_ or $j$-th _row_.
+- $cs_{j,i}$​ is called the $i$-th element of $j$-th _congeneric sequence_.
+- $l$ is _length_, $l \in N$
+- $m$ is _power_, $m \in N$
+
+### from Sequence
+
+Let $alphabet_p$ is [_Alphabet of Partial Sequence function_](../partials_and_congenerics/alphabet/index.md#mathematical-definition)
+
+$$alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+Define
+
+$$congenerics : \{S_p\} \longrightarrow \{CS\}$$
+
+$$congenerics(S_p)(j)(i) = \Bigg\{\begin{array}{l}
+    S_p(i), &   S_p(i)=alphabet_p(S_p)(j) \\
+    -,      &   S_p(i) \in \{-\} \lor S_p(i) \ne alphabet_p(S_p)(j)
+\end{array}$$.
+
+$$\exists congenerics^{-1} : \{CS\} \longrightarrow \{S_p\},$$
+
+$$congenerics^{-1}(CS)(i) = \Bigg\{\begin{array}{l}
+    CS(j)(i), & \exists j \in \{1,...,m\}, CS(j)(i) \notin \{-\} \\
+    -,      &   otherwise
+\end{array}$$
diff --git a/docs/fundamentals/ideas/congeneric_decomposition.md b/docs/fundamentals/ideas/congeneric_decomposition.md
index 878292d3..4bc5b4cf 100644
--- a/docs/fundamentals/ideas/congeneric_decomposition.md
+++ b/docs/fundamentals/ideas/congeneric_decomposition.md
@@ -1,3 +1,156 @@
+---
+hide:
+  - toc
+---
 # Congeneric Decomposition
 
-Coming soon
+Cogeneric decomposition is a method for decomposing symbolic sequences from a systems thinking perspective,
+emphasizing the importance of order. It decomposes a sequence into a tuple of cogeneric sequences,
+each of which consists of equivalent elements at certain positions, while all other positions are empty.
+This reversible process preserves the order of the sequence and allows the original sequence to be fully reconstructed.
+
+The concept of Cogeneric decomposition can be demonstrated using an example:
+
+Let's assume there is a symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT` congeneric decomposition
+could be presented by the following table, where each row is a congeneric sequence and  `-` is an empty position in a congeneric sequence.
+
+``` mermaid
+block-beta
+  columns 36
+  seq1["I"] seq2["N"] seq3["T"] seq4["E"] seq5["L"] seq6["L"] seq7["I"] seq8["G"] seq9["E"] seq10["N"]
+  seq11["C"] seq12["E"] seq13[" "] seq14["I"] seq15["S"] seq16[" "] seq17["T"] seq18["H"] seq19["E"] seq20[" "]
+  seq21["A"] seq22["B"] seq23["I"] seq24["L"] seq25["I"] seq26["T"] seq27["Y"] seq28[" "] seq29["T"] seq30["O"]
+  seq31[" "] seq32["A"] seq33["D"] seq34["A"] seq35["P"] seq36["T"]
+  space:36
+  i1["I"] i2["-"] i3["-"] i4["-"] i5["-"] i6["-"] i7["I"] i8["-"] i9["-"] i10["-"]
+  i11["-"] i12["-"] i13["-"] i14["I"] i15["-"] i16["-"] i17["-"] i18["-"] i19["-"] i20["-"]
+  i21["-"] i22["-"] i23["I"] i24["-"] i25["I"] i26["-"] i27["-"] i28["-"] i29["-"] i30["-"]
+  i31["-"] i32["-"] i33["-"] i34["-"] i35["-"] i36["-"]
+  n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+  n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+  n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"] n30["-"]
+  n31["-"] n32["-"] n33["-"] n34["-"] n35["-"] n36["-"]
+  t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+  t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+  t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"] t30["-"]
+  t31["-"] t32["-"] t33["-"] t34["-"] t35["-"] t36["T"]
+  e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+  e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"]
+  e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"]
+  e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"]
+  l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+  l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+  l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"] l30["-"]
+  l31["-"] l32["-"] l33["-"] l34["-"] l35["-"] l36["-"]
+  g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+  g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+  g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"] g30["-"]
+  g31["-"] g32["-"] g33["-"] g34["-"] g35["-"] g36["-"]
+  c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+  c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+  c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"] c30["-"]
+  c31["-"] c32["-"] c33["-"] c34["-"] c35["-"] c36["-"]
+  sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+  sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+  sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28[" "] sp29["-"] sp30["-"]
+  sp31[" "] sp32["-"] sp33["-"] sp34["-"] sp35["-"] sp36["-"]
+  s1["-"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+  s11["-"] s12["-"] s13["-"] s14["-"] s15["S"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+  s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"] s30["-"]
+  s31["-"] s32["-"] s33["-"] s34["-"] s35["-"] s36["-"]
+  h1["-"] h2["-"] h3["-"] h4["-"] h5["-"] h6["-"] h7["-"] h8["-"] h9["-"] h10["-"]
+  h11["-"] h12["-"] h13["-"] h14["-"] h15["-"] h16["-"] h17["-"] h18["H"] h19["-"] h20["-"]
+  h21["-"] h22["-"] h23["-"] h24["-"] h25["-"] h26["-"] h27["-"] h28["-"] h29["-"] h30["-"]
+  h31["-"] h32["-"] h33["-"] h34["-"] h35["-"] h36["-"]
+  a1["-"] a2["-"] a3["-"] a4["-"] a5["-"] a6["-"] a7["-"] a8["-"] a9["-"] a10["-"]
+  a11["-"] a12["-"] a13["-"] a14["-"] a15["-"] a16["-"] a17["-"] a18["-"] a19["-"] a20["-"]
+  a21["A"] a22["-"] a23["-"] a24["-"] a25["-"] a26["-"] a27["-"] a28["-"] a29["-"] a30["-"]
+  a31["-"] a32["A"] a33["-"] a34["A"] a35["-"] a36["-"]
+  b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+  b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+  b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"] b30["-"]
+  b31["-"] b32["-"] b33["-"] b34["-"] b35["-"] b36["-"]
+  y1["-"] y2["-"] y3["-"] y4["-"] y5["-"] y6["-"] y7["-"] y8["-"] y9["-"] y10["-"]
+  y11["-"] y12["-"] y13["-"] y14["-"] y15["-"] y16["-"] y17["-"] y18["-"] y19["-"] y20["-"]
+  y21["-"] y22["-"] y23["-"] y24["-"] y25["-"] y26["-"] y27["Y"] y28["-"] y29["-"] y30["-"]
+  y31["-"] y32["-"] y33["-"] y34["-"] y35["-"] y36["-"]
+  o1["-"] o2["-"] o3["-"] o4["-"] o5["-"] o6["-"] o7["-"] o8["-"] o9["-"] o10["-"]
+  o11["-"] o12["-"] o13["-"] o14["-"] o15["-"] o16["-"] o17["-"] o18["-"] o19["-"] o20["-"]
+  o21["-"] o22["-"] o23["-"] o24["-"] o25["-"] o26["-"] o27["-"] o28["-"] o29["-"] o30["O"]
+  o31["-"] o32["-"] o33["-"] o34["-"] o35["-"] o36["-"]
+  d1["-"] d2["-"] d3["-"] d4["-"] d5["-"] d6["-"] d7["-"] d8["-"] d9["-"] d10["-"]
+  d11["-"] d12["-"] d13["-"] d14["-"] d15["-"] d16["-"] d17["-"] d18["-"] d19["-"] d20["-"]
+  d21["-"] d22["-"] d23["-"] d24["-"] d25["-"] d26["-"] d27["-"] d28["-"] d29["-"] d30["-"]
+  d31["-"] d32["-"] d33["D"] d34["-"] d35["-"] d36["-"]
+  p1["-"] p2["-"] p3["-"] p4["-"] p5["-"] p6["-"] p7["-"] p8["-"] p9["-"] p10["-"]
+  p11["-"] p12["-"] p13["-"] p14["-"] p15["-"] p16["-"] p17["-"] p18["-"] p19["-"] p20["-"]
+  p21["-"] p22["-"] p23["-"] p24["-"] p25["-"] p26["-"] p27["-"] p28["-"] p29["-"] p30["-"]
+  p31["-"] p32["-"] p33["-"] p34["-"] p35["P"] p36["-"]
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+  class seq1,seq7,seq14,seq23,seq25,i1,i7,i14,i23,i25 c1
+  class seq2,seq10,n2,n10 c2
+  class seq3,seq17,seq26,seq29,seq36,t3,t17,t26,t29,t36 c3
+  class seq4,seq9,seq12,seq19,e4,e9,e12,e19  c4
+  class seq5,seq6,seq24,l5,l6,l24 c5
+  class seq8,g8 c6
+  class seq11,c11 c7
+  class seq13,seq16,seq20,seq28,seq31,sp13,sp16,sp20,sp28,sp31 c8
+  class seq15,s15 c9
+  class seq18,h18 c10
+  class seq21,seq32,seq34,a21,a32,a34 c11
+  class seq22,b22 c12
+  class seq27,y27 c13
+  class seq30,o30 c14
+  class seq33,d33 c15
+  class seq35,p35 c16
+```
+
+Congeneric sequence for `E`
+``` mermaid
+block-beta
+  columns 36
+  e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+  e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"]
+  e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"]
+  e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"]
+```
+
+
+could be a part of multiple symbol sequences that have the same order of `E` element.
+
+While keeping the main idea, the congeneric decomposition could be applied, with a flavor, to any type of special case symbolic sequences, such as Order.
+
+<!-- TODO: Add example of congeneric decomposition code -->
+
+<style>
+.md-typeset table:not([class]) th {
+    min-width: 0 !important;
+}
+
+.md-typeset td:not([class]):not(:last-child), .md-typeset th:not([class]):not(:last-child) {
+    border-right: .05rem solid var(--md-typeset-table-color);
+}
+
+.md-typeset td, .md-typeset th {
+    padding-left: 0.4em !important;
+    padding-right: 0.4em !important;
+    padding-top: 0.1em !important;
+    padding-bottom: 0.1em !important;
+    text-align: center !important;
+}
+</style>
diff --git a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md
index 504d1552..86ba5fb5 100644
--- a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md
+++ b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md
@@ -1,3 +1,40 @@
+---
+hide:
+  - toc
+---
 # Geometric Mean as Alternative to Probability
 
-Coming soon
+At first glance, introducing the concept of [intervals](interval_as_a_basic_information_unit.md) may seem like an unnecessary complication.
+After all, if the ultimate goal is to estimate the probability of a symbol, there are much simpler methods - counting the frequency of occurrence of a symbol relative to the total.
+
+However, this perspective begins to shift when we consider other types of aggregate functions beyond the arithmetic mean.
+One particularly insightful example is the geometric mean of intervals.
+While the arithmetic mean smooths out the "structure" of the data and bring us back to probability
+(since the average interval between identical symbols is simply the inverse of their probability),
+the geometric mean responds to the diversity of intervals in a fundamentally different way.
+
+If the intervals between repeated elements are uniform, the geometric mean and the arithmetic mean will be the same.
+But as the intervals become more irregular — because symbols appear in bursts or clusters — [the geometric mean begins
+to diverge from the arithmetic mean](https://en.wikipedia.org/wiki/AM%E2%80%93GM_inequality).
+This makes it a sensitive indicator of the order within the sequence, not just the frequency.
+
+![AM-GM inequality visual proof](https://upload.wikimedia.org/wikipedia/commons/d/d9/AM_GM_inequality_visual_proof.svg)
+
+*Visual proof of the arithmetic mean - geometric mean inequality. Source: [wikipedia.org](https://en.wikipedia.org/wiki/File:AM_GM_inequality_visual_proof.svg)*
+
+Building on this idea, Former Order Analysis explored the potential of reinterpreting classical probabilistic and information-theoretic measures in terms of these intervals.
+Instead of relying solely on symbol frequencies, it reformulated the measures using the geometric mean instead of probability (arithmetic mean).
+
+\begin{array}{|c|c|}
+\hline
+Entropy & Average \ remoteness \\
+\hline
+H= - \sum_{j=1}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{n_j \log_2 \Delta_{a_j}} & g = \frac{1}{n} * \sum_{j=1}^{m}{n_j \log_2{\Delta_{g_j}}} = \frac{1}{n} * \sum_{j=1}^{m}{\sum_{i=1}^{n_j} \log_2 \Delta_{ij}} \\
+\hline
+\end{array}
+
+*Example of Shennon's Entropy analog - Average remoteness. Where $n$ - seqeunce length, $m$ - alphabet power, $n_j$ - count of element $j$-th, $\Delta_{a_j}$ - average mean of intervals for element $j$-th, $\Delta_{g_j}$ - geometric mean of intervals for element $j$-th, $\Delta_{ij}$ - $i$-th interval for element $j$-thg*
+
+These measures are fine-grained and sensitive to the temporal or spatial order of elements in a sequence.
+Allows us to distinguish between sequences of symbols that may have identical probability distributions
+but differ in the way those symbols are arranged - insight that traditional measures completely miss.
diff --git a/docs/fundamentals/ideas/index.md b/docs/fundamentals/ideas/index.md
index 8c15c86d..b7aecd8b 100644
--- a/docs/fundamentals/ideas/index.md
+++ b/docs/fundamentals/ideas/index.md
@@ -1,3 +1,20 @@
 # Ideas
 
-Coming soon
+Formal Order Analsis is based on five main ideas:
+
+* [Sequence as a Whole Object](sequence_as_a_whole_object.md) -
+Treat a symbolic sequence as a unique whole, focusing on its internal structure and emergent properties.
+
+* [Order as a Property](order_as_a_property.md) -
+By replacing each symbol in a sequence with its index in a dynamic alphabet, we define an "Order" — a new property that separates the content of the sequence from its composition.
+
+* [Congeneric Decomposition](congeneric_decomposition.md) -
+This method breaks a symbolic sequence into layered sub-sequences where each layer isolates a single symbol’s positions,
+preserving the sequence’s internal order and making its structure more analyzable.
+
+* [Interval as a Basic Information Unit](interval_as_a_basic_information_unit.md) -
+Intervals measure the distance between repeated elements, revealing hidden patterns in repetition and spacing.
+
+* [Geometric Mean as Alternative to Probability](geometric_mean_based_characteristics.md) -
+Using measurements based on the geometric mean of intervals instead of the probability of symbols (the arithmetic mean of occurrence)
+allows us to move from recording simple frequency to recording the underlying structure and regularity in the arrangement of symbols.
diff --git a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md
index dba6d4fe..634e1606 100644
--- a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md
+++ b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md
@@ -1,3 +1,54 @@
+---
+hide:
+  - toc
+---
 # Interval as a Basic Information Unit
 
-Coming soon
+Intervals serve as a fundamental unit of information by measuring the number of different
+items, events, or symbols that occur between reseated in a sequence.
+
+The intervals for symbol `A` in the following sequence would be `[3, 3, 1, 1, 2, 1, 1]`
+``` mermaid
+block-beta
+  columns 12
+  s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"]
+  i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["2"]:2 i6["1"]:1 i7["1"]:1
+
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  class s1,s4,s7,s8,s9,s11,s12 c3
+  class i1,i2,i3,i4,i5,i6,i7 c4
+```
+
+In general, a sequence does not necessarily end with the same symbol it begins with.
+To cover all cases, we consider the sequence as a looped sequence representing an infinite pattern with the same characteristics as the original data
+This cyclic approach corresponds to the idea of ​​representativeness heuristic.
+
+The intervals for symbol `C` in the following cycled sequence would be `[3, 9]`
+``` mermaid
+block-beta
+  columns 15
+  s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] space s13["T"] s14["C"]
+  space i1["3"]:3 i2["9"]:10
+  s12 --> s13
+
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  class s2,s5,s14 c3
+  class i1,i2 c4
+```
+
+The circular pattern preserves both the statistical properties and the order of elements.
+Moreover, the average interval length is the inverse of the probability of an event, which directly relates intervals to probability.
+
+\begin{array}{|c|c|c|}
+\hline
+ & \Delta_a  & P \\
+\hline
+A & \frac{3 + 3 + 1 + 1 + 2 + 1 + 1}{7} = \frac{12}{7} \approx 1.7142; & \frac{7}{12} = (\frac{12}{7})^{-1} = \Delta_a^{-1} \\
+\hline
+C & \frac{3 + 9}{2} = \frac{12}{2} = 6 & \frac{2}{12} = \frac{1}{6} = 6^{-1} = \Delta_a^{-1} \\
+\hline
+\end{array}
+
+This makes intervals a crucial informational unit that offers deeper insights into the sequence than individual occurrences alone.
diff --git a/docs/fundamentals/ideas/order_as_a_property.md b/docs/fundamentals/ideas/order_as_a_property.md
index 19deeea3..401d2928 100644
--- a/docs/fundamentals/ideas/order_as_a_property.md
+++ b/docs/fundamentals/ideas/order_as_a_property.md
@@ -1,3 +1,338 @@
+---
+hide:
+  - toc
+---
 # Order as a Property
 
-Coming soon
+Formal order analysis defines a special property of symbolic sequences - an Order.
+The order is a sequence of natural numbers obtained from the original symbolic sequence by replacing each
+of its elements with a natural number corresponding to the index of this element in the alphabet
+sorted by the appearance of the elements in the original sequence [1, 2, 3].
+
+The concept of an Order can be conveniently demonstrated using an example:
+
+
+Let's assume there is a symbolic sequence
+
+``` mermaid
+block-beta
+  columns 36
+  s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+  s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "]
+  s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"]
+  s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"]
+```
+
+
+Find and enumirate the first appearance of each element
+
+``` mermaid
+block-beta
+  columns 36
+  s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+  s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "]
+  s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"]
+  s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"]
+
+  space:36
+
+  i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] space:2 i6_1["6"] space:2 i7_1["7"] space
+  i8_1["8"] space i9_1["9"] space:2 i10_1["10"] space:2
+  i11_1["11"] i12_1["12"] space:4 i13_1["13"]
+  space:2 i14_1["14"] space:2
+  i15_1["15"] space i16_1["16"]
+
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+
+  class s1,i1_1 c1
+  class s2,i2_1 c2
+  class s3,i3_1 c3
+  class s4,i4_1 c4
+  class s5,i5_1 c5
+  class s8,i6_1 c6
+  class s11,i7_1 c7
+  class s13,i8_1 c8
+  class s15,i9_1 c9
+  class s18,i10_1 c10
+  class s21,i11_1 c11
+  class s22,i12_1 c12
+  class s27,i13_1 c13
+  class s30,i14_1 c14
+  class s33,i15_1 c15
+  class s35,i16_1 c16
+
+  s1 --> i1_1
+  s2 --> i2_1
+  s3 --> i3_1
+  s4 --> i4_1
+  s5 --> i5_1
+  s8 --> i6_1
+  s11 --> i7_1
+  s13 --> i8_1
+  s15 --> i9_1
+  s18 --> i10_1
+  s21 --> i11_1
+  s22 --> i12_1
+  s27 --> i13_1
+  s30 --> i14_1
+  s33 --> i15_1
+  s35 --> i16_1
+```
+
+The alphabet (with indexes) for the sequence would be sequence of unique elements:
+
+
+``` mermaid
+block-beta
+  columns 16
+  s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s8["G"]
+  s11["C"] s13[" "] s15["S"] s18["H"]
+  s21["A"] s22["B"] s27["Y"] s30["O"]
+  s33["D"] s35["P"]
+
+  space:16
+
+  i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i6_1["6"] i7_1["7"]
+  i8_1["8"] i9_1["9"] i10_1["10"]
+  i11_1["11"] i12_1["12"] i13_1["13"]
+  i14_1["14"]
+  i15_1["15"] i16_1["16"]
+
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+
+  class s1,i1_1 c1
+  class s2,i2_1 c2
+  class s3,i3_1 c3
+  class s4,i4_1 c4
+  class s5,i5_1 c5
+  class s8,i6_1 c6
+  class s11,i7_1 c7
+  class s13,i8_1 c8
+  class s15,i9_1 c9
+  class s18,i10_1 c10
+  class s21,i11_1 c11
+  class s22,i12_1 c12
+  class s27,i13_1 c13
+  class s30,i14_1 c14
+  class s33,i15_1 c15
+  class s35,i16_1 c16
+
+  s1 --> i1_1
+  s2 --> i2_1
+  s3 --> i3_1
+  s4 --> i4_1
+  s5 --> i5_1
+  s8 --> i6_1
+  s11 --> i7_1
+  s13 --> i8_1
+  s15 --> i9_1
+  s18 --> i10_1
+  s21 --> i11_1
+  s22 --> i12_1
+  s27 --> i13_1
+  s30 --> i14_1
+  s33 --> i15_1
+  s35 --> i16_1
+
+  i1_1 --> s1
+  i2_1 --> s2
+  i3_1 --> s3
+  i4_1 --> s4
+  i5_1 --> s5
+  i6_1 --> s8
+  i7_1 --> s11
+  i8_1 --> s13
+  i9_1 --> s15
+  i10_1 --> s18
+  i11_1 --> s21
+  i12_1 --> s22
+  i13_1 --> s27
+  i14_1 --> s30
+  i15_1 --> s33
+  i16_1 --> s35
+```
+
+
+Determine the order of the sequence by replacing each element of the sequence with its corresponding alphabet index
+
+``` mermaid
+block-beta
+  columns 36
+  s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+  s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "]
+  s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"]
+  s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"]
+
+  space:36
+
+  i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i5_2["5"] i1_2["1"] i6_1["6"] i4_2["4"] i2_2["2"] i7_1["7"] i4_3["4"]
+  i8_1["8"] i1_3["1"] i9_1["9"] i8_2["8"] i3_2["3"] i10_1["10"] i4_4["4"] i8_3["8"]
+  i11_1["11"] i12_1["12"] i1_4["1"] i5_3["5"] i1_5["1"] i3_3["3"] i13_1["13"]
+  i8_4["8"] i3_4["3"] i14_1["14"] i8_5["8"] i11_2["11"]
+  i15_1["15"] i11_3["11"] i16_1["16"] i3_5["3"]
+
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+
+  class s1,s7,s14,s23,s25,i1_1,i1_2,i1_3,i1_4,i1_5 c1
+  class s2,s10,i2_1,i2_2 c2
+  class s3,s17,s26,s29,s36,i3_1,i3_2,i3_3,i3_4,i3_5 c3
+  class s4,s9,s12,s19,i4_1,i4_2,i4_3,i4_4 c4
+  class s5,s6,s24,i5_1,i5_2,i5_3 c5
+  class s8,i6_1 c6
+  class s11,i7_1 c7
+  class s13,s16,s20,s28,s31,i8_1,i8_2,i8_3,i8_4,i8_5 c8
+  class s15,i9_1 c9
+  class s18,i10_1 c10
+  class s21,s32,s34,i11_1,i11_2,i11_3 c11
+  class s22,i12_1 c12
+  class s27,i13_1 c13
+  class s30,i14_1 c14
+  class s33,i15_1 c15
+  class s35,i16_1 c16
+
+
+  s1 --> i1_1
+  s7 --> i1_2
+  s14 --> i1_3
+  s23 --> i1_4
+  s25 --> i1_5
+
+  s2 --> i2_1
+  s10 --> i2_2
+
+  s3 --> i3_1
+  s17 --> i3_2
+  s26 --> i3_3
+  s29 --> i3_4
+  s36 --> i3_5
+
+  s4 --> i4_1
+  s9 --> i4_2
+  s12 --> i4_3
+  s19--> i4_4
+
+  s5 --> i5_1
+  s6 --> i5_2
+  s24 --> i5_3
+
+  s8 --> i6_1
+
+  s11 --> i7_1
+
+  s13 --> i8_1
+  s16 --> i8_2
+  s20 --> i8_3
+  s28 --> i8_4
+  s31 --> i8_5
+
+  s15 --> i9_1
+  s18 --> i10_1
+
+  s21 --> i11_1
+  s32 --> i11_2
+  s34 --> i11_3
+
+  s22 --> i12_1
+  s27 --> i13_1
+  s30 --> i14_1
+  s33 --> i15_1
+  s35 --> i16_1
+```
+
+
+The order of symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT` is
+
+``` mermaid
+block-beta
+  columns 36
+  i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i5_2["5"] i1_2["1"] i6_1["6"] i4_2["4"] i2_2["2"] i7_1["7"] i4_3["4"]
+  i8_1["8"] i1_3["1"] i9_1["9"] i8_2["8"] i3_2["3"] i10_1["10"] i4_4["4"] i8_3["8"]
+  i11_1["11"] i12_1["12"] i1_4["1"] i5_3["5"] i1_5["1"] i3_3["3"] i13_1["13"]
+  i8_4["8"] i3_4["3"] i14_1["14"] i8_5["8"] i11_2["11"]
+  i15_1["15"] i11_3["11"] i16_1["16"] i3_5["3"]
+```
+
+Despite the triviality of the concept Order, it allows us to separate the elements and composition of a sequence and to define the compositional equivalence of different sequences.
+
+Example of sequences with equals orders:
+
+```pyodide exec="on" install="foapy,numpy"
+import foapy
+import numpy as np
+
+seqA = list("INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE")
+seqB = list("1N73LL1G3NC321527H324B1L17Y27024D4P72702CH4NG3")
+orderA = foapy.order(seqA)
+orderB = foapy.order(seqB)
+print("SeqA and SeqB orders are equals -", np.all(orderA == orderB))
+print("Order =", orderA)
+```
+
+<style>
+.md-typeset table:not([class]) th {
+    min-width: 0 !important;
+}
+
+.md-typeset td:not([class]):not(:last-child), .md-typeset th:not([class]):not(:last-child) {
+    border-right: .05rem solid var(--md-typeset-table-color);
+}
+
+.md-typeset td, .md-typeset th {
+    padding-left: 0.4em !important;
+    padding-right: 0.4em !important;
+    padding-top: 0.1em !important;
+    padding-bottom: 0.1em !important;
+    text-align: center !important;
+}
+</style>
+
+# References:
+
+1. Curtis Cooper and Robert E. Kennedy. 1992. Patterns, automata, and Stirling numbers of the second kind. Math. Comput. Educ. 26, 2 (Spring 1992), 120–124.
+2. Gumenjuk A., Kostyshin A., Simonova S. An approach to the research of the structure of linguistic and musical texts. Glottometrics. 2002. № 3. P. 61–89.
+3. (In russian) V.I. Arnold, Complexity of finite sequences of zeros and ones and geometry of finite function spaces: el. print, 2005. http://mms.mathnet.ru/meetings/2005/arnold.pdf
diff --git a/docs/fundamentals/ideas/sequence_as_a_whole_object.md b/docs/fundamentals/ideas/sequence_as_a_whole_object.md
new file mode 100644
index 00000000..3f4b72dd
--- /dev/null
+++ b/docs/fundamentals/ideas/sequence_as_a_whole_object.md
@@ -0,0 +1,26 @@
+# Sequence as a whole object
+
+Symbol sequences are a common model in theoretical and applied science.
+This prevalence is since almost any object of study can be represented as a sequence of elements or events.
+
+However, if we set aside our knowledge of the essence of the elements in a specific case and
+consider the sequence itself as a separate object of study, then it turns out that methods used
+to study them are almost exclusively statistical.
+With the only exception being methods for comparison / alignment of two or more sequences
+(for example, the Levenshtein distance).
+
+None of these methods describe a sequence as a holistic object.
+The Levenshtein distance requires another sequence to compare with the original one, which makes
+the measures of this approach "relative". The probabilistic approach decomposes the sequence into
+elements and calculates their probabilities (frequencies). Thus, the sequence is replaced, as an object of study,
+by a probability distribution. In turn, a specific probability distribution corresponds to an infinite
+number of sequences with a ratio of elements "close" to the one in the original sequence.
+Moments, conditional probabilities, Shannon entropy, and Markov chains allow us to more accurately
+model the object under study, but they still essentially rely on the idea of decomposing a sequence
+into independent elements, ignoring the sequence as a holistic object. Practically all existing approaches
+to the study and description of symbolic sequences originate from the set-theoretic approach.
+
+Formal order analysis is based on the belief that a symbolic sequence can be considered as a holistic object
+with emergent properties, which corresponds to systems thinking. In addition to the distribution
+of elements this method studies arrangement of its components - the internal structure (pattern) of the sequence,
+which determines its uniqueness among others, including those consisting of the same set of elements.
diff --git a/docs/fundamentals/ideas/sequence_as_an_system_object.md b/docs/fundamentals/ideas/sequence_as_an_system_object.md
deleted file mode 100644
index c46c29ec..00000000
--- a/docs/fundamentals/ideas/sequence_as_an_system_object.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Sequence as a System Object
-
-Coming soon
diff --git a/docs/fundamentals/index.md b/docs/fundamentals/index.md
index 5b67ea5f..ab806b84 100644
--- a/docs/fundamentals/index.md
+++ b/docs/fundamentals/index.md
@@ -1,3 +1,13 @@
 # Fundamentals
 
-Coming soon
+Welcome to the Fundamentals section. Here, we introduce the core concepts and mathematical tools that underpin the Formal Order Analysis of sequences and their structure.
+
+- [**Ideas**](./ideas/index.md): This section explores the foundational ideas, motivations, and principles behind Formal Order Analysis.
+
+- [**Order and its measures**](./order/index.md): Learn about the concept of an Order in sequences, including how to quantify and characterize order using measures such as arithmetic mean, geometric mean, and related _interval-based_ characteristics.
+
+- [**Partials and Congenerics**](./partials_and_congenerics/index.md): Discover conceptions of _Partial_ and _Congeneric_ sequences, its Order and other derivatives.
+
+- [**Congeneric Decomposition**](./congeneric_decomposition/index.md): This section details the process and significance of breaking down sequences into congeneric components, providing a framework and measures for deeper structural analysis.
+
+Each section builds on the previous, providing a comprehensive foundation for understanding and applying formal order analysis to a wide range of problems.
diff --git a/docs/fundamentals/objects/alphabet.md b/docs/fundamentals/objects/alphabet.md
deleted file mode 100644
index 0c06aac6..00000000
--- a/docs/fundamentals/objects/alphabet.md
+++ /dev/null
@@ -1,38 +0,0 @@
-# Alphabet
-
-An alphabet is an ordered set of unique elements of a sequence.
-The order of elements in the alphabet is defined by the order of their appearance in the original sequence.
-
-## Mathematical Definition
-
-The alphabet $A$ is defined as an ordered set:
-$$A = \{a_1, a_2, ..., a_m\}$$
-
-Where:
-- $m$ is called a power of the alphabet
-- $a_i \neq a_j$ for all $i \neq j$ where $i,j \in [1..n]$
-- $a_i \in X$ for all $i \in \{1,2,...,m\}$
-- $X$ is an unordered set
-- $=$ is the equivalence relation defined on $X$
-
-## Examples
-
-### Binary Sequence
-For sequence $S = <0,1,1,0,1,0,0,1,1,0>$
-The alphabet is $A = \{0,1\}$
-
-### Musical Chorus
-For sequence $S = <D,Dmaj7,D6,D,D\#dim,Em,A7,Em,A7,Em,A7,Em,A9,A7>$
-The alphabet is $A = \{D,Dmaj7,D6,D,D\#dim,Em,A7,A9\}$
-
-### DNA Sequence
-For sequence $S = <A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C>$
-The alphabet is $A = \{A,T,G,C\}$
-
-### Character Sequence
-For sequence $S = <t,h,e,\ ,q,u,i,c,k,\ ,b,r,o,w,n,\ ,f,o,x,\ ,j,u,m,p,s,\ ,o,v,e,r,\ ,t,h,e,\ ,l,a,z,y,\ ,d,o,g>$
-The alphabet is $A = \{t,h,e,\ ,q,u,i,c,k,b,r,o,w,n,f,x,j,m,p,s,v,l,a,z,y,d,g}$
-
-### Word Sequence
-For sequence $S = <the,\ ,quick,\ ,brown,\ ,fox,\ ,jumps,\ ,over,\ ,the,\ ,lazy,\ ,dog>$
-The alphabet is $A = \{the,\ ,quick, brown, fox, , jumps, over, lazy, dog\}$
diff --git a/docs/fundamentals/objects/index.md b/docs/fundamentals/objects/index.md
deleted file mode 100644
index a314562c..00000000
--- a/docs/fundamentals/objects/index.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Objects
-
-Coming soon
diff --git a/docs/fundamentals/objects/interval.md b/docs/fundamentals/objects/interval.md
deleted file mode 100644
index 473d9beb..00000000
--- a/docs/fundamentals/objects/interval.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Interval
-
-Coming soon
diff --git a/docs/fundamentals/objects/intervals_distribution.md b/docs/fundamentals/objects/intervals_distribution.md
deleted file mode 100644
index d2dd8178..00000000
--- a/docs/fundamentals/objects/intervals_distribution.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Intervals Distribution
-
-Coming soon
diff --git a/docs/fundamentals/objects/intervals_order.md b/docs/fundamentals/objects/intervals_order.md
deleted file mode 100644
index 49c2649a..00000000
--- a/docs/fundamentals/objects/intervals_order.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Intervals Order
-
-Coming soon
diff --git a/docs/fundamentals/objects/intervals_tuple.md b/docs/fundamentals/objects/intervals_tuple.md
deleted file mode 100644
index 7a219029..00000000
--- a/docs/fundamentals/objects/intervals_tuple.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Intervals Tuple
-
-Coming soon
diff --git a/docs/fundamentals/objects/order.md b/docs/fundamentals/objects/order.md
deleted file mode 100644
index 7abd7c2d..00000000
--- a/docs/fundamentals/objects/order.md
+++ /dev/null
@@ -1,3 +0,0 @@
-# Order
-
-Coming soon
diff --git a/docs/fundamentals/order/alphabet.md b/docs/fundamentals/order/alphabet.md
new file mode 100644
index 00000000..bdbeee67
--- /dev/null
+++ b/docs/fundamentals/order/alphabet.md
@@ -0,0 +1,85 @@
+# Alphabet
+
+An alphabet is a m-tuple of unique elements.
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier set_](./carrier_set.md#mathematical-definition)
+
+The _alphabet_ $A$ is a m-tuple with a uniqueness constraint, can be defined:
+
+$$A = <a_1, a_2, ..., a_m>,$$
+
+$$\forall i,j \in \{1, ... ,m\}, i \neq j \implies a_i \neq a_j,$$
+
+$$\forall i \in \{1, ... ,m\} \exists a_i \in X $$
+
+Where:
+
+- $m := |A|$ is called _power_ of the alphabet, $m \in N$
+- $a_i$​ is called the $i$-th _element_ (or coordinate) of the alphabet.
+
+### Alphabet of Sequence
+
+Let $X$ is [_Carrier set_](./carrier_set.md#mathematical-definition)
+
+Let $S$ is [_Sequence_](./sequence.md#mathematical-definition) described as function  $S : \{1,...,n\} \longrightarrow X$
+
+$$alphabet(S) :  \big\{\{1,...,n\} \longrightarrow X \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+$$alphabet(S) = \big<S(i) \big| i \in \{1,...,n\}, \forall k < i,  S(i) \neq S(k)\big>$$
+
+
+Where:
+
+- $m \leq n$ - power of the alphabet is not greater than length of the sequence
+
+## Examples
+
+### Binary Sequence
+A binary sequence `0110100110`
+
+represented as
+
+$X = \{0,1\}$
+
+$A = <0,1>$
+
+### Musical Chorus Sequence
+A musical chorus for `Jingle bell rock`
+
+```
+D                Dmaj7        D6
+Jingle-bell, Jingle-bell, Jingle-bell Rock.
+  D                D#dim
+Jingle-bell swing and
+ Em           A7     Em               A7            Em A7
+Jingle-bell ring. Snowin' and blowin' up bushels of fun.
+Em  A9                  A7
+Now the jingle-hop has begun.
+```
+
+$X = \{A7, A9, D, D6, Dmaj7, D\#dim, Em\}$
+
+$A = <D,Dmaj7,D6,D,D\#dim,Em,A7,A9>$
+
+### DNA Sequence
+A DNA sequence `ATGCTAGCATGCTAGCATGCTAGC`
+
+$X = \{A,C,T,G\}$
+
+$A = <A,T,G,C>$
+
+### English Text Sequence as char sequence
+An English text sentence `the quick brown fox jumps over the lazy dog`
+
+$X = \{\ ,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}$
+
+$A = <t,h,e,\ ,q,u,i,c,k,b,r,o,w,n,f,x,j,m,p,s,v,l,a,z,y,d,g>$
+
+### English Text Sequence as word sequence
+An English text sentence `the quick brown fox jumps over the lazy dog`
+
+$X = \{\ ,quick, fox, brown, the, over, dog, fox, lazy\}$
+
+$A = <the,\ ,quick,brown,fox,jumps,over,lazy,dog>$
diff --git a/docs/fundamentals/order/carrier_set.md b/docs/fundamentals/order/carrier_set.md
new file mode 100644
index 00000000..7dfe8628
--- /dev/null
+++ b/docs/fundamentals/order/carrier_set.md
@@ -0,0 +1,52 @@
+# Carrier set
+
+A _carrier set_ $X$ is a _set_ of all possible elements of a sequence
+
+## Mathematical Definition
+
+Let $X$ be a (possibly infinite) [set](https://en.wikipedia.org/wiki/Set_(mathematics)).
+
+## Examples
+
+### Binary Sequence
+
+A binary sequence `0110100110`
+
+represented as
+
+$X = \{0,1\}$
+
+### Musical Chorus Sequence
+
+A musical chorus for `Jingle bell rock`
+
+```
+D                Dmaj7        D6
+Jingle-bell, Jingle-bell, Jingle-bell Rock.
+  D                D#dim
+Jingle-bell swing and
+ Em           A7     Em               A7            Em A7
+Jingle-bell ring. Snowin' and blowin' up bushels of fun.
+Em  A9                  A7
+Now the jingle-hop has begun.
+```
+
+$X = \{A7, A9, D, D6, Dmaj7, D\#dim, Em\}$
+
+### DNA Sequence
+
+A DNA sequence `ATGCTAGCATGCTAGCATGCTAGC`
+
+$X = \{A,C,G,T\}$
+
+### English Text Sequence as char sequence
+
+An English text sentence `the quick brown fox jumps over the lazy dog`
+
+$X = \{\ ,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z \}$
+
+### English Text Sequence as word sequence
+
+An English text sentence `the quick brown fox jumps over the lazy dog`
+
+$X = \{\ , brown, dog, quick, fox, lazy, over, the\}$
diff --git a/docs/fundamentals/order/characteristics/arithmetic_mean.md b/docs/fundamentals/order/characteristics/arithmetic_mean.md
new file mode 100644
index 00000000..b5fd3ec9
--- /dev/null
+++ b/docs/fundamentals/order/characteristics/arithmetic_mean.md
@@ -0,0 +1,45 @@
+# Arithmetic mean interval
+
+Arithmetic mean interval is an additive measure indifferent to interval ratios.
+The arithmetic mean could be used as a reference for comparing with the geometric mean.
+The geometric mean and the arithmetic mean are equal when all intervals are equal,
+which is true only for the periodic appearance of the elements in the sequence.
+
+## Mathematical Definition
+
+The arithmetic mean interval can be calculated
+
+### from Intervals Chain
+
+Let $IC$ is [_Intervals Chain_](../intervals_chain/index.md#mathematical-definition) described as n-tuple
+
+$$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}$$
+
+$$\Delta_a = \frac{1}{n} \times \sum_{i=1}^{n} \Delta_{i}$$
+
+### from Intervals Distribution
+
+Let $ID$ is [_Intervals Distribution_](../intervals_distribution/index.md#mathematical-definition) described as function
+
+$$ID : \{1,...,n\} \longrightarrow \{1,...,n\}$$
+
+$$\Delta_a(ID) = \frac{\sum_{i=1}^{n} \big(i \times ID(i)\big)}{\sum_{i=1}^{n} ID(i)}$$
+
+### Properties
+
+__With Cycle Bindings, the arithmetic mean interval equals the cardinality of an alphabet__
+
+=== "Intervals chain based"
+    $IC$ produced by [_Cycled Binding_](../intervals_chain/cycled.md#mathematical-definition)
+
+    $\Delta_a = |alphabet(Intervals^{-1}(IC))|$
+
+=== "Sequence based"
+    <!-- is full chain -->
+    Let [Sequence](../sequence.md#mathematical-definition) $S$
+
+    $A=alphabet(S)$ is [Alpabet](../alphabet.md#mathematical-definition)
+
+    $IC = Intervals(S)$ produced by [_Cycled Binding_](../intervals_chain/cycled.md#mathematical-definition)
+
+    then $\Delta_a = |A|$
diff --git a/docs/fundamentals/order/characteristics/average_remoteness.md b/docs/fundamentals/order/characteristics/average_remoteness.md
new file mode 100644
index 00000000..e7dcb9a8
--- /dev/null
+++ b/docs/fundamentals/order/characteristics/average_remoteness.md
@@ -0,0 +1,30 @@
+# Average Remoteness
+
+The Average Remoteness is equivalent of the [geometric mean interval](./geometric_mean.md) on a logarithmic scale.
+It is sensitive to interval ratios and is preferable from [computational point of view](https://en.wikipedia.org/wiki/Log_probability).
+
+## Mathematical Definition
+
+The Average Remoteness interval can be calculated
+
+### from Intervals Chain
+
+Let $IC$ is [_Intervals Chain_](../intervals_chain/index.md#mathematical-definition) described as n-tuple
+
+$$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}$$
+
+$$g = \frac{1}{n} * \sum_{i=1}^{n} \log_2 \Delta_{i}$$
+
+### from Intervals Distribution
+
+Let $ID$ is [_Intervals Distribution_](../intervals_distribution/index.md#mathematical-definition) described as function
+
+$$ID : \{1,...,n\} \longrightarrow \{1,...,n\}$$
+
+$$g(ID) = \frac{\sum_{i=1}^{n} \big(i \times \log_2 ID(i)\big)}{\sum_{i=1}^{n} ID(i)}$$
+
+### Properties
+
+__Geometric mean equals 2 power average remotness__
+
+$$\Delta_g = 2^{g}$$
diff --git a/docs/fundamentals/order/characteristics/depth.md b/docs/fundamentals/order/characteristics/depth.md
new file mode 100644
index 00000000..564f50a4
--- /dev/null
+++ b/docs/fundamentals/order/characteristics/depth.md
@@ -0,0 +1,33 @@
+# Depth
+
+The Depth is equivalent of [Volume](./volume.md) on  on a logarithmic scale.
+It is sensitive to interval ratios and their count (length of the sequence).
+The Depth is better then Volume from [computational point of view](https://en.wikipedia.org/wiki/Log_probability)
+
+## Mathematical Definition
+
+The depth can be calculated
+
+### from Intervals Chain
+
+Let $IC$ is [_Intervals Chain_](../intervals_chain/index.md#mathematical-definition) described as n-tuple
+
+$$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}$$
+
+$$G=\sum_{i=1}^{n} \log_2 \Delta_{i}$$
+
+### from Intervals Distribution
+
+Let $ID$ is [_Intervals Distribution_](../intervals_distribution/index.md#mathematical-definition) described as function
+
+$$ID : \{1,...,n\} \longrightarrow \{1,...,n\}$$
+
+$$V(ID)=\prod_{i=1}^{n} i^{ID(i)}$$
+
+$$G(ID)=\sum_{i=1}^{n} \big(i \times \log_2 ID(i)\big)$$
+
+### Properties
+
+__Volume equals 2 power depth__
+
+$$V = 2^{G}$$
diff --git a/docs/fundamentals/order/characteristics/geometric_mean.md b/docs/fundamentals/order/characteristics/geometric_mean.md
new file mode 100644
index 00000000..df89d5b5
--- /dev/null
+++ b/docs/fundamentals/order/characteristics/geometric_mean.md
@@ -0,0 +1,30 @@
+# Geometric mean interval
+
+The geometric mean interval is a multiplicative measure that is sensitive to interval ratios.
+This property makes it a cornerstone in measuring [Order](../order.md).
+
+## Mathematical Definition
+
+The geometric mean interval can be calculated
+
+### from Intervals Chain
+
+Let $IC$ is [_Intervals Chain_](../intervals_chain/index.md#mathematical-definition) described as n-tuple
+
+$$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}$$
+
+$$\Delta_g=\sqrt[n]{\prod_{i=1}^{n} \Delta_{i}}$$
+
+### from Intervals Distribution
+
+Let $ID$ is [_Intervals Distribution_](../intervals_distribution/index.md#mathematical-definition) described as function
+
+$$ID : \{1,...,n\} \longrightarrow \{1,...,n\}$$
+
+$$\Delta_g(ID)=\sqrt[l]{\prod_{i=1}^{n} i^{ID(i)}}, l = \sum_{i=1}^{n} ID(i)$$
+
+### Properties
+
+__Geometric mean is less or equals arithmetic mean__
+
+$$\Delta_g \le \Delta_a$$
diff --git a/docs/fundamentals/order/characteristics/index.md b/docs/fundamentals/order/characteristics/index.md
new file mode 100644
index 00000000..0408d37d
--- /dev/null
+++ b/docs/fundamentals/order/characteristics/index.md
@@ -0,0 +1,27 @@
+---
+hide:
+  - toc
+---
+# Characteristics
+
+Formal Order Analysis defines several interval-based measures sensitive to the composition of elements in the original sequence.
+All those characteristics are based on two main ideas.
+The first one is - the geometric mean of two numbers depends on the ratio of them.
+The arithmetic mean takes the same value `3` for `1, 5` and `2, 4` pairs, while the geometric mean will be different.
+
+The second is that the intervals we extract from the sequence depend on each other,
+replacing two different neighboring elements in the original sequence will affect two intervals and affect the measure results.
+We need to highlight that intervals, by definition, are measured between the same event appearances
+implicitly encapsulate `information` about the event frequencies and the number of different event types.
+
+In practice, multiplication quickly leads to a stack overflow error. That makes using _linear-scaled_ measures really hard.
+To address that problem, FOA uses _logarithmic-scaled_ measure that replaces multiplication of the intervals with the sum of their logarithms.
+
+All characteristics could be calculated based on [_Intervals Chain_](../intervals_chain/index.md) or [_Intervals Distribution_](../intervals_distribution/index.md).
+The following table provides _Intervals Chain_ based formulas.
+
+| Linear scale | || Logarithmic scale | |
+|------------- |-||-|-----------------|
+| [Arithmetic Mean](arithmetic_mean.md) | $\Delta_a = \frac{1}{n} * \sum_{i=1}^{n} \Delta_{i}$ || | |
+| [Geometric Mean](geometric_mean.md) | $\Delta_g=\sqrt[n]{\prod_{i=1}^{n} \Delta_{i}}$ || $g = \frac{1}{n} * \sum_{i=1}^{n} \log_2 \Delta_{i}$ | [Average Remoteness](average_remoteness.md) |
+| [Volume](volume.md) | $V=\prod_{i=1}^{n} \Delta_{i}$ || $G=\sum_{i=1}^{n} \log_2 \Delta_{i}$ | [Depth](depth.md) |
diff --git a/docs/fundamentals/order/characteristics/volume.md b/docs/fundamentals/order/characteristics/volume.md
new file mode 100644
index 00000000..63243261
--- /dev/null
+++ b/docs/fundamentals/order/characteristics/volume.md
@@ -0,0 +1,32 @@
+# Volume
+
+The Volume is a product of all intervals, which is sensitive to interval ratios and their count.
+The volume value has exponential grow by increasing the interval count (length of the sequence).
+This fact makes it useless in computational models due to overflow error.
+Use [_Depth_](./depth.md) as an equivalent measure on a logarithmic scale.
+
+## Mathematical Definition
+
+The volume can be calculated
+
+### from Intervals Chain
+
+Let $IC$ is [_Intervals Chain_](../intervals_chain/index.md#mathematical-definition) described as n-tuple
+
+$$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}$$
+
+$$V=\prod_{i=1}^{n} \Delta_{i}$$
+
+### from Intervals Distribution
+
+Let $ID$ is [_Intervals Distribution_](../intervals_distribution/index.md#mathematical-definition) described as function
+
+$$ID : \{1,...,n\} \longrightarrow \{1,...,n\}$$
+
+$$V(ID)=\prod_{i=1}^{n} i^{ID(i)}$$
+
+### Properties
+
+__Volume can be calculated base of geometric mean__
+
+$$V = (\Delta_{g})^{n}$$
diff --git a/docs/fundamentals/order/index.md b/docs/fundamentals/order/index.md
new file mode 100644
index 00000000..812e28bc
--- /dev/null
+++ b/docs/fundamentals/order/index.md
@@ -0,0 +1,253 @@
+# Order and its measures
+
+Formal Order Analysis identifies [Order](./order.md) as a sequence's property that could be extracted by replacing elements with their indexes in an [Alphabet](./alphabet.md).
+
+A pair of Alphabet and Order determines a [Sequence](./sequence.md).
+
+=== "Alphabet"
+
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+    s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "]
+    s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"]
+    s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"]
+
+    space:36
+
+    i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] space:2 i6_1["6"] space:2 i7_1["7"] space
+    i8_1["8"] space i9_1["9"] space:2 i10_1["10"] space:2
+    i11_1["11"] i12_1["12"] space:4 i13_1["13"]
+    space:2 i14_1["14"] space:2
+    i15_1["15"] space i16_1["16"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    class s1,i1_1 c1
+    class s2,i2_1 c2
+    class s3,i3_1 c3
+    class s4,i4_1 c4
+    class s5,i5_1 c5
+    class s8,i6_1 c6
+    class s11,i7_1 c7
+    class s13,i8_1 c8
+    class s15,i9_1 c9
+    class s18,i10_1 c10
+    class s21,i11_1 c11
+    class s22,i12_1 c12
+    class s27,i13_1 c13
+    class s30,i14_1 c14
+    class s33,i15_1 c15
+    class s35,i16_1 c16
+
+    s1 --> i1_1
+    s2 --> i2_1
+    s3 --> i3_1
+    s4 --> i4_1
+    s5 --> i5_1
+    s8 --> i6_1
+    s11 --> i7_1
+    s13 --> i8_1
+    s15 --> i9_1
+    s18 --> i10_1
+    s21 --> i11_1
+    s22 --> i12_1
+    s27 --> i13_1
+    s30 --> i14_1
+    s33 --> i15_1
+    s35 --> i16_1
+
+    i1_1  --> s1
+    i2_1  --> s2
+    i3_1  --> s3
+    i4_1  --> s4
+    i5_1  --> s5
+    i6_1  --> s8
+    i7_1  --> s11
+    i8_1  --> s13
+    i9_1  --> s15
+    i10_1 --> s18
+    i11_1 --> s21
+    i12_1 --> s22
+    i13_1 --> s27
+    i14_1 --> s30
+    i15_1 --> s33
+    i16_1 --> s35
+
+    ```
+
+=== "Order"
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+    s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "]
+    s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"]
+    s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"]
+
+    space:36
+
+    i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i5_2["5"] i1_2["1"] i6_1["6"] i4_2["4"] i2_2["2"] i7_1["7"] i4_3["4"]
+    i8_1["8"] i1_3["1"] i9_1["9"] i8_2["8"] i3_2["3"] i10_1["10"] i4_4["4"] i8_3["8"]
+    i11_1["11"] i12_1["12"] i1_4["1"] i5_3["5"] i1_5["1"] i3_3["3"] i13_1["13"]
+    i8_4["8"] i3_4["3"] i14_1["14"] i8_5["8"] i11_2["11"]
+    i15_1["15"] i11_3["11"] i16_1["16"] i3_5["3"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    class s1,s7,s14,s23,s25,i1_1,i1_2,i1_3,i1_4,i1_5 c1
+    class s2,s10,i2_1,i2_2 c2
+    class s3,s17,s26,s29,s36,i3_1,i3_2,i3_3,i3_4,i3_5 c3
+    class s4,s9,s12,s19,i4_1,i4_2,i4_3,i4_4 c4
+    class s5,s6,s24,i5_1,i5_2,i5_3 c5
+    class s8,i6_1 c6
+    class s11,i7_1 c7
+    class s13,s16,s20,s28,s31,i8_1,i8_2,i8_3,i8_4,i8_5 c8
+    class s15,i9_1 c9
+    class s18,i10_1 c10
+    class s21,s32,s34,i11_1,i11_2,i11_3 c11
+    class s22,i12_1 c12
+    class s27,i13_1 c13
+    class s30,i14_1 c14
+    class s33,i15_1 c15
+    class s35,i16_1 c16
+
+
+    s1 --> i1_1
+    s7 --> i1_2
+    s14 --> i1_3
+    s23 --> i1_4
+    s25 --> i1_5
+
+    s2 --> i2_1
+    s10 --> i2_2
+
+    s3 --> i3_1
+    s17 --> i3_2
+    s26 --> i3_3
+    s29 --> i3_4
+    s36 --> i3_5
+
+    s4 --> i4_1
+    s9 --> i4_2
+    s12 --> i4_3
+    s19--> i4_4
+
+    s5 --> i5_1
+    s6 --> i5_2
+    s24 --> i5_3
+
+    s8 --> i6_1
+
+    s11 --> i7_1
+
+    s13 --> i8_1
+    s16 --> i8_2
+    s20 --> i8_3
+    s28 --> i8_4
+    s31 --> i8_5
+
+    s15 --> i9_1
+    s18 --> i10_1
+
+    s21 --> i11_1
+    s32 --> i11_2
+    s34 --> i11_3
+
+    s22 --> i12_1
+    s27 --> i13_1
+    s30 --> i14_1
+    s33 --> i15_1
+    s35 --> i16_1
+    ```
+
+---
+
+Studying the Order FOA developed methods of measuring the Order that are very sensitive to the composition of the elements in a sequence.
+
+The following diagrams give a hand in understanding how the Objects and Methods defined in FOA relates to each other.
+
+=== "Order as a property"
+    ```mermaid
+    flowchart TB
+        Start@{ shape: sm-circ, label: "" }-- Sequence -->fork1@{ shape: sm-circ, label: "" }
+        fork1-- Sequence -->alphabet
+        alphabet-- Alphabet -->OA@{ shape: sm-circ }
+        fork1-- Sequence -->order
+        order-- Order -->OA
+        OA-- Sequence -->End@{ shape: sm-circ }
+
+        click alphabet "./alphabet" "Alphabet"
+        click order "./order" "Order"
+    ```
+
+=== "Interval-based characteristics"
+    ```mermaid
+    flowchart TB
+        Start@{ shape: sm-circ, label: "" }-- Sequence -->fork1@{ shape: sm-circ, label: "" }
+        fork1-- Sequence -->alphabet
+        alphabet-- Alphabet -->OA@{ shape: cross-circ }
+        order-- Order -->OA@{ shape: sm-circ }
+        OA-- Sequence -->End@{ shape: sm-circ }
+
+        fork1@{ shape: sm-circ, label: "" }-- Sequence -->intervals
+        intervals-- Intervals Chain -->fork@{ shape: sm-circ, label: "" }
+        fork@{ shape: sm-circ, label: "" }-- Intervals Chain -->inverseIntervals@{ label: "intervals⁻¹" }
+        fork@{ shape: sm-circ, label: "" }-- Intervals Chain -->distribution
+        inverseIntervals@{ label: "intervals⁻¹" }-- Reconstructed Sequence --> order
+        distribution-- Intervals Distribution --> M@{ shape: sm-circ, label: "order + alphabet" }
+        M --> da@{ label: "Δa" }
+        M --> dg@{ label: "Δg" }
+        M --> g@{ label: "g" }
+        M --> V@{ label: "V" }
+        M --> G@{ label: "G" }
+        da -- float --> EndMeasureda@{ shape: sm-circ }
+        dg -- float --> EndMeasuredg@{ shape: sm-circ }
+        g -- float --> EndMeasureg@{ shape: sm-circ }
+        V -- int --> EndMeasureV@{ shape: sm-circ }
+        G -- float --> EndMeasureG@{ shape: sm-circ }
+
+
+        click alphabet "./alphabet" "Alphabet"
+        click order "./order" "Order"
+        click intervals "./intervals_chain/#define-intervals-chain" "Intervals"
+        click inverseIntervals "./intervals_chain/#define-intervals-chain" "Intervals⁻"
+        click distribution "./intervals_distribution" "Distribution"
+        click distribution "./intervals_distribution" "Distribution"
+        click dg "./characteristics/geometric_mean" "Geomteric mean"
+        click da "./characteristics/arithmetic_mean" "Arithmetic mean"
+        click g "./characteristics/average_remoteness" "Average remoteness"
+        click G "./characteristics/depth" "Depth"
+        click V "./characteristics/volume" "Volume"
+    ```
diff --git a/docs/fundamentals/order/intervals_chain/bounded.md b/docs/fundamentals/order/intervals_chain/bounded.md
new file mode 100644
index 00000000..60ea4cb8
--- /dev/null
+++ b/docs/fundamentals/order/intervals_chain/bounded.md
@@ -0,0 +1,247 @@
+# Bounded Intervals Chain
+
+A _bounded intervals chain_ is an [_intervals chain_](index.md) produced with _Bounded Binding_.
+_Bounded Binding_ treats a sequence as a finite and uses $0$ and $n+1$ positions (depedns of _Iterator_ direction)
+as _corresponding position_ anytime there is no `next` matching element.
+The approach simplifies implementation functions and enables obtaining _binding direction_ based on a given _bounded intervals chain_ due to its specific properties.
+This comes with a cost of the intervals' consistency that depends on _binding direction_ and leads to different measure values.
+_Bounded Binding_ identifies _Start_ and _End_ directions.
+
+
+=== "$Start$ binding"
+
+    ``` mermaid
+    block-beta
+      columns 8
+      p0["0"]        p1["1"] space:3         p5["5"]                p6["n"] space
+      inf["⊥"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] space
+      e0["0 = Iterator(1)"]:2 space:6
+      space      t1["1 = Iterator(5)"]:5         space:2
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5 c4
+      class inf,t1,e0 c4a
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+=== "$End$ binding"
+
+    ``` mermaid
+    block-beta
+      columns 8
+      space        p1["1"] space:3         p5["5"]                p6["n"] p7["n + 1"]
+      space s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] sup["⊥"]
+      space      t1["Iterator(1) = 5"]:5         space:2
+      space:5 e0["Iterator(5) = n+1"]:3
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5 c4
+      class sup,t1,e0 c4a
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier set_](../carrier_set.md#mathematical-definition)
+
+Let $S$ is [_Sequence_](../sequence.md#mathematical-definition)  length of $n$ described as function $S : \{1,...,n\} \longrightarrow X$
+
+Let $Binding$ is [Binding](./index.md#define-bindings)
+
+### Define Bindings
+
+=== "$Start$ binding"
+    Define a set of terminal values - $\bot = \{0\}$
+
+    Let $R : \{1,...,n\} \longrightarrow \{1,...,n\} \cup \bot,$ is a corresponding _references_
+
+    Define
+
+    $$Iterator : \big\{ S \big\} \longrightarrow \big\{ R \big\},$$
+
+    $$Iterator(S)(i) = \Bigg\{\begin{array}{l} max \big\{j \in \{1,...,i\}\big|S(j) = S(i) \land j \ne i \big\} & if \ exists \\ 0 & otherwise  \end{array}$$
+
+    $$Start = <Iterator, \bot> \in \{Binding\}$$
+
+    $$\exists Start^{-1} = <Iterator^{-1}, \bot>  \in \{Binding^{-1}\},$$
+
+    $$Iterator^{-1} : \big\{  R \big\} \longrightarrow \big\{  \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\}$$
+
+    $$Iterator^{-1}(R)(i) = \Bigg\{\begin{array}{l} \Big|<j \in \{1,...i\} | R(j) \in \bot >\Big|, & \ R(i) \in \bot \\ Iterator^{-1}(R, R(i)), &  R(i) \in \{1,...,n\}   \end{array}$$
+
+    $$Iterator(S) = Iterator(Iterator^{-1}(Iterator(S)))$$
+
+
+=== "$End$ binding"
+    Define a set of terminal values - $\bot = \{n+1\}$
+
+    Let $R : \{1,...,n\} \longrightarrow \{1,...,n\} \cup \bot,$ is a corresponding _references_
+
+    Define
+
+    $$Iterator : \big\{ S \big\} \longrightarrow \big\{ R \big\},$$
+
+    $$Iterator(S)(i) = \Bigg\{\begin{array}{l} min \big\{j \in \{i,...,n\}\big|S(j) = S(i) \land j \ne i \big\} & if \ exists \\ n+1 & otherwise  \end{array}$$
+
+    $$End = <Iterator, \bot> \in \{Binding\}$$
+
+    $$\exists End^{-1} = <Iterator^{-1}, \bot>  \in \{Binding^{-1}\},$$
+
+    $$Iterator^{-1} : \big\{  R \big\} \longrightarrow \big\{  \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\}$$
+
+    $$Iterator^{-1}(R)(i) = \Bigg\{\begin{array}{l} \Big|<j \in \{i,...n\} | R(j) \in \bot >\Big|, & \ R(i) \in \bot \\ Iterator^{-1}(R, R(i)), &  R(i) \in \{1,...,n\} \end{array}$$
+
+    $$Iterator(S) = Iterator(Iterator^{-1}(Iterator(S)))$$
+
+---
+
+### Define Intervals Chain
+
+=== "$Start$ binding"
+
+    Define
+
+    $$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\},$$
+
+    $$\exists \ Intervals : \big\{S\}  \longrightarrow \big\{ IC \big\},$$
+
+    $$Intervals(S)(i) = | i - Iterator(S)(i)|,$$
+
+    $$\exists \ Follow : \big\{ IC \big\}  \longrightarrow \big\{ R \big\},$$
+
+    $$Follow(IC)(i) = i - IC(i),$$
+
+    $$\exists Intervals^{-1} : \big\{ IC \big\}  \longrightarrow \big\{ \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\},$$
+
+    $$Intervals^{-1}(IC)(i) = Iterator^{-1}(Follow(IC))(i),$$
+
+    $$Intervals(S) = Intervals(Intervals^{-1}(Intervals(S)))$$
+
+    $$\exists Trace : \big\{ IC \big\} \longrightarrow \big\{ R \big\}$$
+
+    $$Trace(IC)(i) = \Bigg\{\begin{array}{l} i, & \ i \in \bot \\ Trace\big(IC\big)\big(Follow(IC, i)\big) &  i \in \{1,...,n\} \end{array}$$
+
+
+    Where:
+
+    - $n := |IC|$ is called _length_ of the _intervals chained_, $n \in N$
+    - $\Delta_i$​ is called the $i$-th _element_ (or interval) of the _intervals chained_
+
+
+=== "$End$ binding"
+
+    Define
+
+    $$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\},$$
+
+    $$\exists \ Intervals : \big\{S\}  \longrightarrow \big\{ IC \big\},$$
+
+    $$Intervals(S)(i) = | i - Iterator(S)(i)|,$$
+
+    $$\exists \ Follow : \big\{ IC \big\}  \longrightarrow \big\{ R \big\},$$
+
+    $$Follow(IC)(i) = i + IC(i),$$
+
+    $$\exists Intervals^{-1} : \big\{ IC \big\}  \longrightarrow \big\{ \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\},$$
+
+    $$Intervals^{-1}(IC)(i) = Iterator^{-1}(Follow(IC))(i),$$
+
+    $$Intervals(S) = Intervals(Intervals^{-1}(Intervals(S)))$$
+
+    $$\exists Trace : \big\{ IC \big\} \longrightarrow \big\{ R \big\}$$
+
+    $$Trace(IC)(i) = \Bigg\{\begin{array}{l} i, & \ i \in \bot \\ Trace\big(IC\big)\big(Follow(IC, i)\big) &  i \in \{1,...,n\} \end{array}$$
+
+
+    Where:
+
+    - $n := |IC|$ is called _length_ of the _intervals chained_, $n \in N$
+    - $\Delta_i$​ is called the $i$-th _element_ (or interval) of the _intervals chained_
+
+
+---
+
+#### Special properties
+
+_Intervals chain_ $IC$ have been calculated with _Bounded binding_ have a special properties
+
+=== "$Start$ binding"
+
+    $$IC(1) = 1$$
+
+    $$IC(i) \le i | \forall i \in \{1,...,n\}$$
+
+    $$Follow(IC)(i) <> Follow(IC)(j) \lor Follow(IC)(i) \in \bot | \forall i != j$$
+
+    $$Trace(IC)(i) = 0 | \forall i \in \{1,...,n\}$$
+
+=== "$End$ binding"
+
+    $$IC(n) = 1$$
+
+    $$IC(i) \le n - i | \forall i \in \{1,...,n\}$$
+
+    $$Follow(IC)(i) <> Follow(IC)(j) \lor Follow(IC)(i) \in \bot | \forall i != j$$
+
+    $$Trace(IC)(i) = n + 1 | \forall i \in \{1,...,n\}$$
+
+
+---
+
+Let $B = \{Start, End\} \subset \{Binding\}$ is set of _Bounded Binding_
diff --git a/docs/fundamentals/order/intervals_chain/cycled.md b/docs/fundamentals/order/intervals_chain/cycled.md
new file mode 100644
index 00000000..64b56c74
--- /dev/null
+++ b/docs/fundamentals/order/intervals_chain/cycled.md
@@ -0,0 +1,264 @@
+# Cycled Intervals Chain
+
+A _cycled intervals chain_ is an [_intervals chain_](index.md) produced with _Cycled Binding_.
+_Cycled Binding_ treats a sequence as a subsequence representing an infinite sequence.
+
+The approach alignged with [Representativeness heuristic](https://en.wikipedia.org/wiki/Representativeness_heuristic) idea,
+connects FOA with [Necklace](https://en.wikipedia.org/wiki/Necklace_(combinatorics)) problem and
+makes intervals based measures indeferent to _binding direction_.
+
+_Cycled Binding_ identifies _Start_ and _End_ directions.
+
+_Cycled Binding_ extends the sequence by copying itself as a prefix and suffix.
+This is enough to mock it as a cycled sequence (also known as a periodic sequence or an orbit) and
+use the prefix and suffix to find the corresponding position for the element in edge cases.
+
+
+=== "$Start$ binding"
+
+    ``` mermaid
+    block-beta
+      columns 18
+      pomn["-n+1"]    space:4  p00["0"]    p01["1"] space:4                                     p06["n"] p07["n + 1"]  space:4 p02n["n + n"]
+      p1["A"] p2["C"] p3["T"] p4["C"] p5["A"] p6["G"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] f1["A"] f2["C"] f3["T"] f4["C"] f5["A"] f6["G"]
+      space:5 ia1["2"]:2 space:11
+      space:4 ic1["4"]:4 space:10
+      space:3 it1["6"]:6 space:9
+      space:6 ig1["6"]:6 space:6
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,ia1 c2
+      class p5 c2a
+      class s2,ic1 c4
+      class p4 c4a
+      class s3,it1 c6
+      class p3 c6a
+      class s6,ig1 c14
+      class p6 c14a
+      class p1,p2,p3,p4,p5,p6,p7,p8,p9,f1,f2,f3,f4,f5,f6,f7,f8,f9 imaginary
+      class pomn,p00,p01,p06,p07,p02n position
+
+    ```
+
+=== "$End$ binding"
+
+    ``` mermaid
+    block-beta
+      columns 18
+      pomn["-n+1"]    space:4  p00["0"]    p01["1"] space:4                                     p06["n"] p07["n + 1"]  space:4 p02n["n + n"]
+      p1["A"] p2["C"] p3["T"] p4["C"] p5["A"] p6["G"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] f1["A"] f2["C"] f3["T"] f4["C"] f5["A"] f6["G"]
+      space:10 ia1["2"]:2 space:6
+      space:9 ic1["4"]:4 space:5
+      space:8 it1["6"]:6 space:5
+      space:10 ig1["6"]:6 space:1
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s5,ia1 c2
+      class f1 c2a
+      class s4,ic1 c4
+      class f2 c4a
+      class s3,it1 c6
+      class f3 c6a
+      class s6,ig1 c14
+      class f6 c14a
+      class p1,p2,p3,p4,p5,p6,p7,p8,p9,f1,f2,f3,f4,f5,f6,f7,f8,f9 imaginary
+      class pomn,p00,p01,p06,p07,p02n position
+
+    ```
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier set_](../carrier_set.md#mathematical-definition)
+
+Let $S$ is [_Sequence_](../sequence.md#mathematical-definition)  length of $n$ described as function $S : \{1,...,n\} \longrightarrow X$
+
+Let $Binding$ is [Binding](./index.md#define-bindings)
+
+Let $S_{cycled} : \big\{ \{1,...,n\} \longrightarrow X \big\} \longrightarrow \big\{ Z \longrightarrow X \big\}$ is a cycled sequence
+
+$$S_{cycled}(S)(i) = S\big( i - n \times \lfloor ( i - 1) \div n \rfloor \big)$$
+
+
+### Define Bindings
+
+=== "$Start$ binding"
+    Define a set of terminal values - $\bot = \{-n+1,...,0\}$
+
+    Let $R : \{1,...,n\} \longrightarrow \{1,...,n\} \cup \bot,$ is a corresponding _references_
+
+    Define
+
+    $$Iterator : \big\{ S \big\} \longrightarrow \big\{ R \big\},$$
+
+    $$Iterator(S)(i) = max \big\{j \in \{-n+1,...,i\}\big|S_{cycled}(j) = S(i) \land j \ne i \big\} $$
+
+    $$Start = <Iterator, \bot> \in \{Binding\}$$
+
+    $$\exists Start^{-1} = <Iterator^{-1}, \bot>  \in \{Binding^{-1}\},$$
+
+    $$Iterator^{-1} : \big\{  R \big\} \longrightarrow \big\{  \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\}$$
+
+    $$Iterator^{-1}(R)(i) = \Bigg\{\begin{array}{l} \Big|<j \in \{1,...i\} | R(j) \in \bot >\Big|, & \ R(i) \in \bot \\ Iterator^{-1}(R, R(i)), &  R(i) \in \{1,...,n\}   \end{array}$$
+
+    $$Iterator(S) = Iterator(Iterator^{-1}(Iterator(S)))$$
+
+
+=== "$End$ binding"
+    Define a set of terminal values - $\bot = \{n+1,...,2n\}$
+
+    Let $R : \{1,...,n\} \longrightarrow \{1,...,n\} \cup \bot,$ is a corresponding _references_
+
+    Define
+
+    $$Iterator : \big\{ S \big\} \longrightarrow \big\{ R \big\},$$
+
+    $$Iterator(S)(i) = min \big\{j \in \{i,...,2n\}\big|S_{cycled}(j) = S(i) \land j \ne i \big\}$$
+
+    $$End = <Iterator, \bot> \in \{Binding\}$$
+
+    $$\exists End^{-1} = <Iterator^{-1}, \bot>  \in \{Binding^{-1}\},$$
+
+    $$Iterator^{-1} : \big\{  R \big\} \longrightarrow \big\{  \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\}$$
+
+    $$Iterator^{-1}(R)(i) = \Bigg\{\begin{array}{l} \Big|<j \in \{i,...n\} | R(j) \in \bot >\Big|, & \ R(i) \in \bot \\ Iterator^{-1}(R, R(i)), &  R(i) \in \{1,...,n\} \end{array}$$
+
+    $$Iterator(S) = Iterator(Iterator^{-1}(Iterator(S)))$$
+
+---
+
+### Define Intervals Chain
+
+=== "$Start$ binding"
+
+    Define
+
+    $$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\},$$
+
+    $$\exists \ Intervals : \big\{S\}  \longrightarrow \big\{ IC \big\},$$
+
+    $$Intervals(S)(i) = | i - Iterator(S)(i)|,$$
+
+    $$\exists \ Follow : \big\{ IC \big\}  \longrightarrow \big\{ R \big\},$$
+
+    $$Follow(IC)(i) = i - IC(i),$$
+
+    $$\exists Intervals^{-1} : \big\{ IC \big\}  \longrightarrow \big\{ \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\},$$
+
+    $$Intervals^{-1}(IC)(i) = Iterator^{-1}(Follow(IC))(i),$$
+
+    $$Intervals(S) = Intervals(Intervals^{-1}(Intervals(S)))$$
+
+    $$\exists Trace : \big\{ IC \big\} \longrightarrow \big\{ R \big\}$$
+
+    $$Trace(IC)(i) = \Bigg\{\begin{array}{l} i, & \ i \in \bot \\ Trace\big(IC\big)\big(Follow(IC, i)\big) &  i \in \{1,...,n\} \end{array}$$
+
+
+    Where:
+
+    - $n := |IC|$ is called _length_ of the _intervals chained_, $n \in N$
+    - $\Delta_i$​ is called the $i$-th _element_ (or interval) of the _intervals chained_
+
+
+=== "$End$ binding"
+
+    Define
+
+    $$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\},$$
+
+    $$\exists \ Intervals : \big\{S\}  \longrightarrow \big\{ IC \big\},$$
+
+    $$Intervals(S)(i) = | i - Iterator(S)(i)|,$$
+
+    $$\exists \ Follow : \big\{ IC \big\}  \longrightarrow \big\{ R \big\},$$
+
+    $$Follow(IC)(i) = i + IC(i),$$
+
+    $$\exists Intervals^{-1} : \big\{ IC \big\}  \longrightarrow \big\{ \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\},$$
+
+    $$Intervals^{-1}(IC)(i) = Iterator^{-1}(Follow(IC))(i),$$
+
+    $$Intervals(S) = Intervals(Intervals^{-1}(Intervals(S)))$$
+
+    $$\exists Trace : \big\{ IC \big\} \longrightarrow \big\{ R \big\}$$
+
+    $$Trace(IC)(i) = \Bigg\{\begin{array}{l} i, & \ i \in \bot \\ Trace\big(IC\big)\big(Follow(IC, i)\big) &  i \in \{1,...,n\} \end{array}$$
+
+
+    Where:
+
+    - $n := |IC|$ is called _length_ of the _intervals chained_, $n \in N$
+    - $\Delta_i$​ is called the $i$-th _element_ (or interval) of the _intervals chained_
+
+
+---
+
+#### Special properties
+
+_Intervals chain_ $IC$ have been calculated with _Bounded binding_ have a special properties
+
+=== "$Start$ binding"
+
+    $$IC(i) \le n | \forall i \in \{1,...,n\}$$
+
+    $$Follow(IC)(i) <> Follow(IC)(j) | \forall i != j$$
+
+    $$j < 1 \land j = Trace(IC)(n + j) | \forall i \in \{1,...,n\} \exists j = Trace(IC)(i)$$
+
+=== "$End$ binding"
+
+    $$IC(i) \le n | \forall i \in \{1,...,n\}$$
+
+    $$Follow(IC)(i) <> Follow(IC)(j) | \forall i != j$$
+
+    $$j > n+1 \land j = Trace(IC)(j - n) | \forall i \in \{1,...,n\} \exists j = Trace(IC)(i)$$
+
+
+---
+
+Let $B = \{Start, End\} \subset \{Binding\}$ is set of _Cycled Binding_
diff --git a/docs/fundamentals/order/intervals_chain/index.md b/docs/fundamentals/order/intervals_chain/index.md
new file mode 100644
index 00000000..32ec0ee4
--- /dev/null
+++ b/docs/fundamentals/order/intervals_chain/index.md
@@ -0,0 +1,483 @@
+# Intervals Chain
+
+An _intervals chain_ is an n-tuple of natural numbers that represents the distance between equal elements in a sequence,
+if and only if there is an operation that takes as an input _current_ $(index, interval)$ and returns the next _index_.
+The operation makes the intervals chained to each other, which is a required condition for the existence reverse function
+that restores by _intervals chain_ a sequence with the same with the original sequence [order](../order.md).
+
+The idea of _intervals chain_ is easy to explain by a concrete example:
+
+
+=== "From a sequence"
+
+    ``` mermaid
+    block-beta
+        columns 8
+        space        p1["1"] space:3         p5["5"]                p6["n"] space
+        space s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] space
+
+        classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+        classDef position fill:#fff,color:#000,stroke-width:0px;
+        class inf,sup imaginary
+        class p0,p1,p5,p6,p7 position
+
+        classDef c1 fill:#ff7f0e,color:#fff;
+        classDef c2 fill:#ffbb78,color:#000;
+        classDef c2a fill:#ffbb788a,color:#000;
+        classDef c3 fill:#2ca02c,color:#fff;
+        classDef c4 fill:#98df8a,color:#000;
+        classDef c4a fill:#98df8a8a,color:#000;
+        classDef c5 fill:#d62728,color:#fff;
+        classDef c6 fill:#ff9896,color:#000;
+        classDef c6a fill:#ff98968a,color:#000;
+        classDef c7 fill:#9467bd,color:#fff;
+        classDef c8 fill:#c5b0d5,color:#000;
+        classDef c9 fill:#8c564b,color:#fff;
+        classDef c10 fill:#c49c94,color:#000;
+        classDef c11 fill:#e377c2,color:#fff;
+        classDef c12 fill:#f7b6d2,color:#000;
+        classDef c13 fill:#bcbd22,color:#fff;
+        classDef c14 fill:#dbdb8d,color:#000;
+        classDef c14a fill:#dbdb8d8a,color:#000;
+        classDef c15 fill:#17becf,color:#fff;
+        classDef c16 fill:#9edae5,color:#000;
+    ```
+
+    Let there be a sequence length of $n=6$ (indexed from 1 to n=6)
+
+=== "with a binding"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]        p1["1"] space:3         p5["5"]                p6["6"]
+      inf["⊥"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"]
+      e0["0 = Iterator(1)"]:2 space:5
+      space      t1["1 = Iterator(5)"]:5         space
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5 c4
+      class inf,t1,e0 c4a
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Define _Binding_ as a pair of an _Iterator_ function, that seeks a corresponding referenced element, and
+    set of _terminate states_, to determine the interval when there is no matching element in the sequence.
+
+    In the example, _Iterator_ seeks a position of a previous equivalent element as a matching reference.
+    If there is no such element, it returns $0$. That means _terminate states_ set is $\bot = {0}$.
+    This particular method is called - _Start binding_.
+
+=== "get correspoding indexes"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      inf["⊥"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"]
+      space i1["0"] i2["0"] i3["0"] i4["2"] i5["1"] i6["0"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 c4
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    With _Binding_ we can get a sequence of corresponding indexes.
+    For all first appearances of the elements it would be $0$,
+    and for `C` at position `4` corresponding index would be `2`, for `A` at position `5` - `1`.
+
+=== "and calculate intervals chain"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space i1["0"] i2["0"] i3["0"] i4["2"] i5["1"] i6["0"]
+      space s1["1"] s2["2"] s3["3"] s4["2"] s5["4"] s6["6"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 c4
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Calculated interval by $Intervals$ function is the [absolute value](https://en.wikipedia.org/wiki/Absolute_value)
+    of difference the _corresponding index_ and the _index_. For example, for index `5` interval would be $|5-1|=4$
+
+
+---
+
+There should exists an inverse function that restores by _interval chain_ a sequence with the original [order](../order.md)
+
+=== "From an interval chain"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["1"] s2["2"] s3["3"] s4["2"] s5["4"] s6["6"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Let there be an _interval chain_.
+
+=== "calculate correspoding indexes"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["1"] s2["2"] s3["3"] s4["2"] s5["4"] s6["6"]
+      space i1["0"] i2["0"] i3["0"] i4["2"] i5["1"] i6["0"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    The $Intervals^{-1}$ function for calculating _corresponding index_ based on the current _index_ and _interval_ is closly coupled
+    with direction in selected _Binding_. The function has to be defined for each particular _Binding_ method.
+    Whether it is possible to obtain direction of _Binding_ given an _interval chain_, and whether there are two equivalent _interval chains_
+    obtained by different _Binding_ methods, are open questions that need to be investigated.
+
+    In the example $Intervals^{-1}(IC)(i) = IC(i)-i$
+
+=== "and use binding⁻¹"
+
+    ``` mermaid
+    block-beta
+      columns 6
+      p1["1"] space:3         p5["5"]                p6["6"]
+      s1["0"] s2["0"] s3["0"] s4["2"] s5["1"] s6["0"]
+      e0["Iterator⁻¹(1) = 1"] space:5
+      t1["Iterator⁻¹(5) = 1"]:5         space
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5 c4
+      class inf,t1,e0 c4a
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    For each _Binding_  should exists $Binding^{-1}$ with an $Iterator^{-1}$ that allows
+    to relabel elements of _interval chain_ with a unique number of the traversed path that it belongs to.
+
+    $$Iterator^{-1}(P)(i) = \Bigg\{\begin{array}{l} \big|<j \in \{1,...i\} | P(j)=0 >\big| & if \ P(i)=0 \\ Iterator^{-1}(P, P(i)) & otherwise   \end{array}$$
+
+=== "to reconstruct sequence"
+
+    ``` mermaid
+    block-beta
+      columns 6
+      p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      i1["0"] i2["0"] i3["0"] i4["2"] i5["1"] i6["0"]
+      s1["1"] s2["2"] s3["3"] s4["2"] s5["1"] s6["4"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 c4
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    The reconstructed sequence is not equal to the original one, but it preserves the same [order](../order.md) of elements
+    and its _intervals chain_ would be equal to the _interval chain_ of the original sequence.
+
+    ``` mermaid
+    block-beta
+      columns 6
+      p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      s1["1"] s2["2"] s3["3"] s4["2"] s5["1"] s6["4"]
+      i1["A"] i2["C"] i3["T"] i4["C"] i5["A"] i6["G"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 c4
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+---
+
+Formal order analysis identifies two groups of _Bindings_:
+
+- [Bounded](./bounded.md) - consider a sequence to be finite and bounded. Operates with the minimum data required to determine _intervals chains_ and _sequence_.
+- [Cycled](./cycled.md) - treats a sequence as a subsequence representing an infinite sequence. Connects FOA with the fundamental ideas underlying statistics and probability theory.
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier set_](../carrier_set.md#mathematical-definition)
+
+Let $S$ is [_Sequence_](../sequence.md#mathematical-definition)  length of $n$ described as function $S : \{1,...,n\} \longrightarrow X$
+
+### Define Bindings
+
+Define a set of terminal values - $\bot \subset N \setminus \{1,..,n\}$
+
+Let $R : \{1,...,n\} \longrightarrow \{1,...,n\} \cup \bot,$ is a corresponding _references_
+
+Define
+
+$$Iterator : \big\{  S \big\} \longrightarrow \big\{ R \big\},$$
+
+$$Binding = <Iterator, \bot>$$
+
+$$\exists Binding^{-1} = <Iterator^{-1}, \bot>,$$
+
+$$Iterator^{-1} : \big\{  R \big\} \longrightarrow \big\{  \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\}$$
+
+$$Iterator(S) = Iterator(Iterator^{-1}(Iterator(S)))$$
+
+### Define Intervals Chain
+
+as n-tuple of natural numbers
+
+$$IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\},$$
+
+if and only if
+
+$$\exists \ Intervals : \big\{Binding\big\} \times \big\{S\}  \longrightarrow \big\{ IC \big\},$$
+
+$$Intervals(<iterator, \bot>, S)(i) = | i - iterator(S)(i)|,$$
+
+$$\exists \ Intervals^{-1} : \big\{Binding^{-1}\big\} \times \big\{ IC \big\}  \longrightarrow \big\{ \{1,...,n\} \longrightarrow \{1,...,m\} | m \leq n \big\},$$
+
+$$Intervals(b, S) = Intervals(b, Intervals^{-1}(b^{-1}, Intervals(b, S)))$$
+
+$$\exists \ Follow : \big\{Binding\big\} \times \big\{ IC \big\}  \longrightarrow \big\{ R \big\},$$
+
+$$Follow(b, IC)(i) <> Follow(b, IC)(j) \lor Follow(b, IC)(i) \in \bot | \forall i != j $$
+
+$$\exists \ Trace : \big\{Binding\big\} \times \big\{ IC \big\} \longrightarrow \big\{ R \big\}$$
+
+$$Trace(<iterator, \bot>, IC)(i) = \Bigg\{\begin{array}{l} i, & \ i \in \bot \\ Trace\big(Follow(IC, i)\big) &  i \in \{1,...,n\} \end{array}$$
+
+$$Trace(IC)(i) \in \bot | \forall i \in \{1,...,n\}$$
+
+Where:
+
+- $n := |IC|$ is called _length_ of the _intervals chained_, $n \in N$
+- $\Delta_i$​ is called the $i$-th _element_ (or interval) of the _intervals chained_
diff --git a/docs/fundamentals/order/intervals_distribution/index.md b/docs/fundamentals/order/intervals_distribution/index.md
new file mode 100644
index 00000000..105a978a
--- /dev/null
+++ b/docs/fundamentals/order/intervals_distribution/index.md
@@ -0,0 +1,119 @@
+# Intervals Distribution
+
+An _intervals distribution_ is an n-tuple of natural numbers where the index represents the interval length and the value is a count of its appearances in the _interval chain_.
+
+
+=== "From an interval chain"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["1"] s2["2"] s3["3"] s4["2"] s5["4"] s6["6"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Let there be an _interval chain_.
+
+=== "calculate intervals distribution"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["1"] s2["2"] s3["3"] s4["2"] s5["4"] s6["6"]
+      space:7
+      space i1["1"] i2["2"] i3["1"] i4["1"] i5["0"] i6["1"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+      s1 --> i1
+      s2 --> i2
+      s3 --> i3
+      s4 --> i2
+      s5 --> i4
+      s6 --> i6
+
+
+    ```
+
+---
+
+_Intervals distribution_ used as an input data in calculating [characteristics](../characteristics/index.md).
+While characteristics could be calculated based on the _itervals chain_ _intervals distribution_ highlights that intervals themselves are enough to measure the [order](../order.md) of a sequence
+and intervals connectivity in _intervals chain_ does not affect measure values.
+Whether it is possible in general to reconstruct distinctly an _interval chain_ by the given _interval distribution_ is an open question.
+
+
+
+_Interval distribution_ is useful in comparing intervals produced from the same sequence with different [_Binding_](../intervals_chain/index.md#define-bindings).
+In the interest of studying how _intervals_ depend on _Binding direction_ for [_Bounded Binding_](../intervals_chain/bounded.md) FOA introduce two operations on distributions:
+
+- [Lossy](./lossy.md) - takes two _intervals distribution_ and produce new one only with _intervals_ exists in both distributions.
+- [Redundant](./redundant.md) - extends _intervals distribution_ `A` with _intervals_ that appears only in _intervals distrubution_ `B`.
+
+## Mathematical Definition
+
+Let $IC$ is [_Interval Chain_](../intervals_chain/index.md#define-intervals-chain) length of $n$ described as function $IC : \{1,...,n\} \longrightarrow  \{1,...,n\}$
+
+Define
+
+$$ID : \big\{  IC \big\} \longrightarrow \big\{ \{1,...,n\} \longrightarrow  N_0 \big\},$$
+
+$$ID(IC)(i) = \Big| \big\{ j \in \{1,...,n\} | IC(j) = i \big\} \Big|$$
diff --git a/docs/fundamentals/order/intervals_distribution/lossy.md b/docs/fundamentals/order/intervals_distribution/lossy.md
new file mode 100644
index 00000000..caa6c6f7
--- /dev/null
+++ b/docs/fundamentals/order/intervals_distribution/lossy.md
@@ -0,0 +1,64 @@
+# Lossy Intervals Distribution
+
+A _lossy intervals distribution_ is an aggregation of two _interval distributions_ having only intervals existing in both.
+
+_Lossy Interval Distribution_ is an intersection of two distributions.
+
+Mostly, this distribution is used to solve measure dependence on _Binding direction_ with [_Bounded binding_](../intervals_chain/bounded.md).
+In that case, this would be equivalent to excluding the first/last intervals from the distribution.
+
+For example, there are 2 distributions for _Bounded Binding_ - one uses _Start binding direction_ and the other  _End binding direction_.
+
+``` mermaid
+block-beta
+    columns 8
+    p0["0"]        p1["1"] space:3         p5["5"]                p6["6"] p7["7"]
+    inf["⊥"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] sup["⊥"]
+    space es0["Start(1) = 1"]:1 ts1["Start(5)=4"]:5 space
+    space      te1["End(1) = 4"]:4 ee0["End(5) = 2"]:2 space
+
+    classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+    classDef position fill:#fff,color:#000,stroke-width:0px;
+    class inf,sup imaginary
+    class p0,p1,p5,p6,p7 position
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c2a fill:#ffbb788a,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c4a fill:#98df8a8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c6a fill:#ff98968a,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c14a fill:#dbdb8d8a,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    class s1,s5 c4
+    class inf,sup,te1,ee0,ts1,es0 c4a
+    class pomn,p00,p01,p06,p07,p02n position
+```
+
+_Lossy Interval Distribution_ will contain only interval `4` as it exists in both distributions.
+
+
+
+
+## Mathematical Definition
+
+Let $ID$ is [_Intervals distribution_](../intervals_distribution/index.md#mathematical-definition)
+
+Define _Lossy Interval Distribution_
+
+$$LID: \{ID\} \times \{ID\} \longrightarrow \{ID\}$$
+
+$$LID(ID_1, ID_2)(i) = min \{ID_1(i), ID_2(i) \}$$
diff --git a/docs/fundamentals/order/intervals_distribution/redundant.md b/docs/fundamentals/order/intervals_distribution/redundant.md
new file mode 100644
index 00000000..3f45a07c
--- /dev/null
+++ b/docs/fundamentals/order/intervals_distribution/redundant.md
@@ -0,0 +1,62 @@
+# Redundant Intervals Distribution
+
+A _redundant intervals distribution_ is an aggregation of two _interval distributions_ having all intervals from the first one and intervals existing only the second.
+
+_Redundant Interval Distribution_ is a union of two distributions.
+
+
+Mostly, this distribution is used to solve measure dependence on _Binding direction_ with [_Bounded binding_](../intervals_chain/bounded.md).
+In that case, this would be equivalent to counting both intervals - first for _Start binding direction_ and _End binding direction_.
+
+For example, there are 2 distributions for _Bounded Binding_ - one uses _Start binding direction_ and the other  _End binding direction_.
+
+``` mermaid
+block-beta
+    columns 8
+    p0["0"]        p1["1"] space:3         p5["5"]                p6["6"] p7["7"]
+    inf["⊥"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] sup["⊥"]
+    space es0["Start(1) = 1"]:1 ts1["Start(5)=4"]:5 space
+    space      te1["End(1) = 4"]:4 ee0["End(5) = 2"]:2 space
+
+    classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+    classDef position fill:#fff,color:#000,stroke-width:0px;
+    class inf,sup imaginary
+    class p0,p1,p5,p6,p7 position
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c2a fill:#ffbb788a,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c4a fill:#98df8a8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c6a fill:#ff98968a,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c14a fill:#dbdb8d8a,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    class s1,s5 c4
+    class inf,sup,te1,ee0,ts1,es0 c4a
+    class pomn,p00,p01,p06,p07,p02n position
+```
+
+_Redundant Interval Distribution_ will include all intervals `[1, 4, 2]`.
+
+## Mathematical Definition
+
+Let $ID$ is [_Intervals distribution_](../intervals_distribution/index.md#mathematical-definition)
+
+Define _Redundunt Interval Distribution_
+
+$$RID: \{ID\} \times \{ID\} \longrightarrow \{ID\}$$
+
+$$RID(ID_1, ID_2)(i) = max \{ID_1(i), ID_2(i) \}$$
diff --git a/docs/fundamentals/order/order.md b/docs/fundamentals/order/order.md
new file mode 100644
index 00000000..dbd558a6
--- /dev/null
+++ b/docs/fundamentals/order/order.md
@@ -0,0 +1,150 @@
+# Order
+
+An _order_ is a n-tuple of natural numbers starting from `1` with `max+1` constraint.
+
+## Mathematical Definition
+
+The _order_ $O$ is defined as a n-tuple with additional constraints:
+
+$$O = <o_1, o_2, ..., o_n>,$$
+
+$$\forall i \in \{1, ..., n\} \exists o_i \in N $$
+
+$$ o_1 = 1, $$
+
+$$\forall i \in \{1, ..., n\}, o_i \leq max(o_1, ..., o_{i-1}) + 1$$
+
+Where:
+
+- $n := |O|$ is called _length_ of the order, $n \in N$
+- $o_i$​ is called the $i$-th _element_ (or coordinate) of the order
+
+### Order of Sequence
+
+Let $X$ is [_Carrier set_](./carrier_set.md#mathematical-definition)
+
+Let $S$ is [_Sequence_](./sequence.md#mathematical-definition) described as function  $S : \{1,...,n\} \longrightarrow X$
+
+Let $alphabet$ is [_Alphabet function_](./alphabet.md#alphabet-of-sequence)
+
+$$alphabet : \big\{\{1,...,n\} \longrightarrow X \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+Define
+
+$$ordert(S) :  \big\{\{1,...,n\} \longrightarrow X \big\} \longrightarrow \big\{\{1,...,n\} \longrightarrow \{1,...,n\} \big\}$$
+
+$$A = alphabet(S)$$
+
+$$order(S)(i) = j \big| j \in \{1,...,n\}, S(i)=A(j)$$
+
+### Order product Alphabet
+
+Let $X$ is a [_Carrier set_](./carrier_set.md)
+
+Let $A$ is a [_Aphabet_](./alphabet.md) $A : \{1, ..., m\} \longrightarrow X,$
+
+Let $S$ is a [_Sequenece_](./sequence.md) $S : \{1, ..., n\} \longrightarrow X,$
+
+the following equations are true
+
+$$O = order(S),$$
+
+$$A = alphabet(S),$$
+
+$$S = ( O \odot A),$$
+
+$$S(i) = A(O(i))$$
+
+
+
+
+## Examples
+
+### Valid order
+
+``` mermaid
+block-beta
+  columns 36
+  i1["1"] i2["2"] i3["3"] i4["2"] i5["4"] i6["2"]
+  i7["5"] i8["2"] i9["6"] i10["2"] i11["7"] i12["2"]
+  i13["1"] i14["2"] i15["8"] i16["2"] i17["9"]
+
+  classDef red fill:#d62728,color:#000;
+
+```
+
+### Invalid order - Start different from `1`
+
+
+``` mermaid
+block-beta
+  columns 36
+  i1["2"] i2["2"] i3["3"] i4["2"] i5["4"] i6["2"]
+  i7["5"] i8["2"] i9["6"] i10["2"] i11["7"] i12["2"]
+  i13["1"] i14["2"] i15["8"] i16["2"] i17["9"]
+
+  classDef red fill:#d62728,color:#000;
+
+  class i1 red
+```
+
+### Invalid order - Contains elements not in `N`
+
+``` mermaid
+block-beta
+  columns 36
+  i1["1"] i2["2"] i3["3"] i4["2"] i5["4"] i6["2"]
+  i7["5"] i8["2"] i9["6"] i10["T"] i11["7"] i12["2"]
+  i13["1"] i14["-2"] i15["8"] i16["2"] i17["9"]
+
+  classDef red fill:#d62728,color:#000;
+
+  class i10,i14 red
+```
+
+### Invalid order - Violates `max + 1` contstraint
+
+``` mermaid
+block-beta
+  columns 36
+  i1["1"] i2["2"] i3["3"] i4["2"] i5["4"] i6["2"]
+  i7["5"] i8["2"] i9["6"] i10["2"] i11["7"] i12["2"]
+  i13["1"] i14["9"] i15["8"] i16["2"] i17["9"]
+
+  classDef red fill:#d62728,color:#000;
+
+  class i14 red
+```
+
+### Binary Sequence
+A binary sequence `0110100110`
+
+represented as
+
+$O = <1, 2, 2, 1, 2, 1, 1, 2, 2, 1>$
+
+### Musical Chorus Sequence
+A musical chorus for `Jingle bell rock`
+
+```
+D                Dmaj7        D6
+Jingle-bell, Jingle-bell, Jingle-bell Rock.
+  D                D#dim
+Jingle-bell swing and
+ Em           A7     Em               A7            Em A7
+Jingle-bell ring. Snowin' and blowin' up bushels of fun.
+Em  A9                  A7
+Now the jingle-hop has begun.
+```
+
+$O = <1, 2, 3, 1, 4, 5, 6, 5, 6, 7, 5, 8, 6>$
+
+### DNA Sequence
+A DNA sequence `ATGCTAGCATGCTAGCATGCTAGC`
+
+$O = <1, 2, 3, 4, 2, 1, 3, 4, 1, 2, 3, 4, 2, 1, 3, 4, 1, 2, 3, 4, 2, 1, 3, 4>$
+
+### English Text Sequence as word sequence
+An English text sentence `the quick brown fox jumps over the lazy dog`
+
+$O = <1, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 1, 2, 8, 2, 9>$
diff --git a/docs/fundamentals/objects/sequence.md b/docs/fundamentals/order/sequence.md
similarity index 71%
rename from docs/fundamentals/objects/sequence.md
rename to docs/fundamentals/order/sequence.md
index 9724cc74..7203ee78 100644
--- a/docs/fundamentals/objects/sequence.md
+++ b/docs/fundamentals/order/sequence.md
@@ -1,20 +1,28 @@
 # Sequence
 
 A sequence is a fundamental concept in formal order analysis that is defined as a finite, enumerated collection of objects in which repetitions are allowed and order matters.
-It is the basic object for analyzing patterns, relationships, and structural properties in ordered data.
+This is the basic object for analyzing patterns, relationships, and structural properties in ordered data.
 
 ## Mathematical Definition
 
-A sequence $S$ is as tuple:
+Let $X$ is [_Carrier set_](./carrier_set.md#mathematical-definition)
 
-$$S = <s_1, s_2, ..., s_n>$$
+A _sequence_ $S$ is a n-tuple defined as
+
+$$S = <s_1, s_2, ..., s_n>,$$
+
+$$\forall i \in \{1, ..., n\} \exists s_i \in X$$
 
 where:
 
-- $n$ is the length of the sequence
-- $s_i \in X$ for all $i \in \{1,2,...,n\}$
-- $X$ is an unordered set
-- $=$ is the equivalence relation defined on $X$
+- $s_i$​ is called the $i$-th _element_ (or coordinate) of the sequence.
+- $n := |S|$ is _length_, $n \in N$
+
+The _sequence_ $S$ can be also defined as a function
+
+$$S : \{1, ..., n\} \longrightarrow X,$$
+
+$$S(i)=s_i | i \in \{1, ..., n\}$$
 
 ## Examples
 
@@ -41,23 +49,27 @@ Em  A9                  A7
 Now the jingle-hop has begun.
 ```
 
-$X = \{D, D6, D\#dim, A7, A9, Dmaj7, Em\}$
+$X = \{A7, A9, D, D6, Dmaj7, D\#dim, Em\}$
+
 $S = <D,Dmaj7,D6,D,D\#dim,Em,A7,Em,A7,Em,A7,Em,A9,A7>$
 
 ### DNA Sequence
 A DNA sequence `ATGCTAGCATGCTAGCATGCTAGC`
 
 $X = \{A,C,T,G\}$
+
 $S = <A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C>$
 
 ### English Text Sequence as char sequence
 An English text sentence `the quick brown fox jumps over the lazy dog`
 
 $X = \{\ ,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}$
+
 $S = <t,h,e,\ ,q,u,i,c,k,\ ,b,r,o,w,n,\ ,f,o,x,\ ,j,u,m,p,s,\ ,o,v,e,r,\ ,t,h,e,\ ,l,a,z,y,\ ,d,o,g>$
 
 ### English Text Sequence as word sequence
 An English text sentence `the quick brown fox jumps over the lazy dog`
 
 $X = \{\ ,quick, fox, brown, the, over, dog, fox, lazy\}$
+
 $S = <the,\ ,quick,\ ,brown,\ ,fox,\ ,jumps,\ ,over,\ ,the,\ ,lazy,\ ,dog>$
diff --git a/docs/fundamentals/partials_and_congenerics/alphabet/congeneric.md b/docs/fundamentals/partials_and_congenerics/alphabet/congeneric.md
new file mode 100644
index 00000000..2b87cbd6
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/alphabet/congeneric.md
@@ -0,0 +1,17 @@
+# Alphabet of Congeneric Sequence
+
+[_Congenric sequence_](../sequence/congeneric.md#mathematical-definition) is defined on _Partial carrier set_ having only one _non-empty_ element.
+That means that _Alphabet of Congeneric sequence_ is a tuple of one element. The alphabet power is 1.
+
+## Mathematical Definition
+
+
+Let $S_{c|e}$ is [_Congeneric Sequence of e element_](../sequence/congeneric.md#mathematical-definition)
+
+Let $alphabet_p$ is [_Alphabet of Partial sequence_](../alphabet/index.md#mathematical-definition)
+
+Then [_Alphabet_](../../order/alphabet.md#mathematical-definition)
+
+$$A=alphabet_p(S_{c|e}) = <e>,$$
+
+$$|A| = 1$$
diff --git a/docs/fundamentals/partials_and_congenerics/alphabet/index.md b/docs/fundamentals/partials_and_congenerics/alphabet/index.md
new file mode 100644
index 00000000..3d03add1
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/alphabet/index.md
@@ -0,0 +1,22 @@
+# Alphabet of Partial Sequence
+
+_Alphabet_ of [_Partial sequence_](../sequence/index.md#mathematical-definition) is just an [_Alphabet_](../../order/alphabet.md#mathematical-definition).
+The `-` _empty element_ is not counted as a unique element and is not included in the alphabet.
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier Set_](../../order/carrier_set.md#mathematical-definition)
+
+Let $X_{-}$ is [_Partial Carrier Set_](../carrier_set.md#mathematical-definition)
+
+$$X \subset X_{-}$$
+
+Let $-$ is [_Empty element_](../carrier_set.md#mathematical-definition) of _Partial Carrier Set_
+
+Let $S_p$ is [_Partial Sequence_](../sequence/index.md#mathematical-definition) described as function  $S : \{1,...,l\} \longrightarrow X_{-}$
+
+Define _Alphabet of partial sequence_ function
+
+$$alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+$$alphabet_p(S_p) = \big<S_p(i) \big| i \in \{1,...,l\}, \forall k < i, S_p(i) \notin \{-\} \land S_p(i) \neq S_p(k) \big>$$
diff --git a/docs/fundamentals/partials_and_congenerics/carrier_set.md b/docs/fundamentals/partials_and_congenerics/carrier_set.md
new file mode 100644
index 00000000..7c6856ee
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/carrier_set.md
@@ -0,0 +1,13 @@
+# Partial carrier set
+
+A _Partial carrier set_ $X_{-}$ is a [_carrier set_](../order/carrier_set.md) extended with special `-` empty element.
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier set_](../order/carrier_set.md#mathematical-definition)
+
+Let $- \notin X$
+
+Define _Partial carrier set_
+
+$$X_{-} = X \cup \{-\}$$
diff --git a/docs/fundamentals/partials_and_congenerics/characteristics.md b/docs/fundamentals/partials_and_congenerics/characteristics.md
new file mode 100644
index 00000000..6189d4d7
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/characteristics.md
@@ -0,0 +1,10 @@
+---
+hide:
+  - toc
+---
+# Characteristics for Partials
+
+All [interval-based characteristics](../order/characteristics/index.md) are valid of _partials_ and _congenerics_.
+The measures could be calculated based on [_Partial Intervals Chain_](./intervals_chain/index.md) or [_Partial Intervals Distribution_](./intervals_distribution.md) using the
+relevant formulas for the characteristic. In case of using _Partial Intervals Chain_ pay attention that for partial $n$ is the number of _non-empty_ elements and differs from
+tuple length $l$, while for _full_ sequences and their derivatives $n$ and $l$ are the same.
diff --git a/docs/fundamentals/partials_and_congenerics/index.md b/docs/fundamentals/partials_and_congenerics/index.md
new file mode 100644
index 00000000..5d983516
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/index.md
@@ -0,0 +1,105 @@
+# Partials and congenerics
+
+_Partial sequence_ is a sequence with skips (empty positions, spaces). FOA uses the special symbol `-` as _empty_ element.
+_Partial sequence_ where all _non-empty_ elements are equals _Congeneric sequence_.
+
+=== "Sequence"
+
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+    s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "]
+    s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"]
+    s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#ffffff
+
+    ```
+
+=== "Partial"
+
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+    s11["C"] s12["E"] s13[" "] s14["-"] s15["-"] s16[" "] s17["T"] s18["-"] s19["-"] s20[" "]
+    s21["-"] s22["B"] s23["-"] s24["L"] s25["-"] s26["T"] s27["-"] s28["-"] s29["T"] s30["-"]
+    s31["-"] s32["-"] s33["D"] s34["-"] s35["P"] s36["T"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#ffffff
+
+    class s14,s15,s18,s19,s21,s23,s25,s27,s28,s30,s31,s32,s34 skip
+    ```
+
+=== "Congeneric"
+
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["-"] s2["-"] s3["T"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+    s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["T"] s18["-"] s19["-"] s20["-"]
+    s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["T"] s27["-"] s28["-"] s29["T"] s30["-"]
+    s31["-"] s32["-"] s33["-"] s34["-"] s35["-"] s36["T"]
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#ffffff
+
+    class s1,s2,s4,s5,s6,s7,s8,s9,s10 skip
+    class s11,s12,s13,s14,s15,s16,s18,s19,s20 skip
+    class s21,s22,s23,s24,s25,s27,s28,s30 skip
+    class s31,s32,s33,s34,s35 skip
+    ```
+
+Every operation and statement described in [_Order and its measures_](../order/index.md) that can be applied to _Sequence_ are also valid for Partial sequence, keeping in mind that only non-empty elements are involved.
+Read the following documentation to clarify the differences that appear for  Partial [_Sequence_](./sequence/index.md), [_Order_](./order/index.md), and [_Intervals Chain_](./intervals_chain/index.md).
diff --git a/docs/fundamentals/partials_and_congenerics/intervals_chain/congeneric.md b/docs/fundamentals/partials_and_congenerics/intervals_chain/congeneric.md
new file mode 100644
index 00000000..20dd17eb
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/intervals_chain/congeneric.md
@@ -0,0 +1,63 @@
+# Congeneric Intervals Chain
+
+A _Congeneric intervals chain_ is an [_Partial Interval Chain_](./index.md) where all _non-empty_ elements are part of the one trace path (trace to the same [terminal value](../../order/intervals_chain/index.md)).
+
+``` mermaid
+block-beta
+columns 29
+
+i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+s1["-"] s2["-"] s3["3"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["14"] s18["-"] s19["-"] s20["-"]
+s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["9"] s27["-"] s28["-"] s29["3"]
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s1,s2,s4,s5,s6,s7,s8,s9,s10 skip
+class s11,s12,s13,s14,s15,s16,s18,s19,s20 skip
+class s21,s22,s23,s24,s25,s27,s28,s30 skip
+class s31,s32,s33,s34,s35 skip
+
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+```
+
+## Mathematical Definition
+
+Let $-$ is [_empty value_](../carrier_set.md#mathematical-definition)
+
+Let $IC_p$ is [_Partial interval chain_](index.md#define-partial-intervals-chain) $IC_p : \{1, ..., l\} \longrightarrow \{1,...,l\} \cup \{-\},$
+
+Let $Trace_p$ is [_trace function of partial interval chain_](index.md#define-partial-intervals-chain)
+
+$$Trace_p : \big\{Binding_p\big\} \times \big\{ IC_p \big\} \longrightarrow \big\{ R_p \big\},$$
+
+$IC_p$ is called $IC_c$ _Congeneric interval chain_ if
+
+$$trace = Trace_p(b_p, IC_P)$$
+
+$$trace(i) = trace(j) \bigg| \forall i \ne j, IC_{p}(i) \notin \{-\} \land  IC_{p}(j) \notin \{-\}$$
diff --git a/docs/fundamentals/partials_and_congenerics/intervals_chain/index.md b/docs/fundamentals/partials_and_congenerics/intervals_chain/index.md
new file mode 100644
index 00000000..e8e1615e
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/intervals_chain/index.md
@@ -0,0 +1,579 @@
+# Partial Intervals Chain
+
+_Partial Intervals Chain_ is an [_Intervals chain_](../../order/intervals_chain/index.md#mathematical-definition) that could contains `-` [_empty_](../carrier_set.md#mathematical-definition) elements. All statements valid for an _intervals chain elements_  should be valid for _non-empty_ elements of _partial intervals chain_
+
+The idea of _partial intervals chain_ is easy to explain by a concrete example:
+
+
+=== "From a partial sequence"
+
+    ``` mermaid
+    block-beta
+        columns 8
+        space        p1["1"] space:3         p5["5"]                p6["l"] space
+        space s1["-"] s2["C"] s3["T"] s4["C"] s5["-"] s6["G"] space
+
+        classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+        classDef position fill:#fff,color:#000,stroke-width:0px;
+        class inf,sup imaginary
+        class p0,p1,p5,p6,p7 position
+
+        classDef skip fill:#ffffff
+
+        class s1,s5 skip
+
+        classDef c1 fill:#ff7f0e,color:#fff;
+        classDef c2 fill:#ffbb78,color:#000;
+        classDef c2a fill:#ffbb788a,color:#000;
+        classDef c3 fill:#2ca02c,color:#fff;
+        classDef c4 fill:#98df8a,color:#000;
+        classDef c4a fill:#98df8a8a,color:#000;
+        classDef c5 fill:#d62728,color:#fff;
+        classDef c6 fill:#ff9896,color:#000;
+        classDef c6a fill:#ff98968a,color:#000;
+        classDef c7 fill:#9467bd,color:#fff;
+        classDef c8 fill:#c5b0d5,color:#000;
+        classDef c9 fill:#8c564b,color:#fff;
+        classDef c10 fill:#c49c94,color:#000;
+        classDef c11 fill:#e377c2,color:#fff;
+        classDef c12 fill:#f7b6d2,color:#000;
+        classDef c13 fill:#bcbd22,color:#fff;
+        classDef c14 fill:#dbdb8d,color:#000;
+        classDef c14a fill:#dbdb8d8a,color:#000;
+        classDef c15 fill:#17becf,color:#fff;
+        classDef c16 fill:#9edae5,color:#000;
+    ```
+
+    Let there be a partial sequence length of $l=6$ (indexed from 1 to l=6)
+
+=== "with a binding"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]        p1["1"] p2["2"] space  p4["4"]       p5["5"]                p6["6"]
+      inf["⊥"] s1["-"] s2["C"] s3["T"] s4["C"] s5["-"] s6["G"]
+      e0["0 = Iteratorₚ(2)"]:3 space:4
+      space:2      t1["2 = Iteratorₚ(4)"]:3         space:2
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+      class s1,s5 skip
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s2,s4 c4
+      class inf,t1,e0 c4a
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Define $Binding_p$ as a pair of an $Iterator_p$ function, that seeks a corresponding referenced element, and
+    set of _terminate states_, to determine the interval when there is no matching element in the sequence.
+    For `-` _empty_ elements, we do not search for matching elements - just put `-` instead of matching value.
+
+=== "get correspoding indexes"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      inf["⊥"] s1["-"] s2["C"] s3["T"] s4["C"] s5["-"] s6["G"]
+      space i1["-"] i2["0"] i3["0"] i4["2"] i5["-"] i6["0"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+      class s1,s5 skip
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 skip
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    With $Binding_p$ we can get a sequence of corresponding indexes.
+    For all first appearances of the elements it would be $0$,
+    and for `C` at position `4` corresponding index would be `2`.
+
+=== "and calculate partial intervals chain"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space i1["-"] i2["0"] i3["0"] i4["2"] i5["-"] i6["0"]
+      space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 skip
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Calculated interval by $Intervals_p$ function is the [absolute value](https://en.wikipedia.org/wiki/Absolute_value)
+    of difference the _corresponding index_ and the _index_. For example, for index `4` interval would be $|4-2|=2$
+
+
+---
+
+There should exists an inverse function that restores by _interval chain_ a sequence with the original [order](../order/index.md)
+
+=== "From a partial interval chain"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+      class s1,s5 skip
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Let there be a _partial interval chain_.
+
+=== "calculate correspoding indexes"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      p0["0"]  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]
+      space i1["-"] i2["0"] i3["0"] i4["2"] i5["-"] i6["0"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+      class s1,s5,i1,i5 skip
+
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    The $Intervals_p^{-1}$ function for calculating _corresponding index_ based on the current _index_ and _interval_ is closly coupled
+    with direction in selected _Binding_. For `-` _empty_ element it just put `-` instead of calculating the index
+
+=== "and use binding⁻¹"
+
+    ``` mermaid
+    block-beta
+      columns 6
+      p1["1"] p2["2"] space p4["4"]         p5["5"]                p6["6"]
+      s1["-"] s2["0"] s3["0"] s4["2"] s5["-"] s6["0"]
+      e0["Iteratorₚ⁻¹(2) = 2"]:2 space:4
+      space t1["Iteratorₚ⁻¹(4) = 2"]:3         space:2
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+      class s1,s5,i1,i5 skip
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s2,s4 c4
+      class inf,t1,e0 c4a
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    For each $Binding_p$  should exists $Binding_p^{-1}$ with an $Iterator_p^{-1}$ that allows
+    to relabel elements of _interval chain_ with a unique number of the traversed path that it belongs to.
+
+=== "to partial sequence"
+
+    ``` mermaid
+    block-beta
+      columns 6
+      p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      i1["-"] i2["0"] i3["0"] i4["2"] i5["-"] i6["0"]
+      s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["4"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+      class s1,s5,i1,i5 skip
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    The reconstructed partial sequence is not equal to the original one, but it preserves the same [partial order](../order/index.md) of elements
+    and its _partial intervals chain_ would be equal to the _partial interval chain_ of the original sequence.
+
+    ``` mermaid
+    block-beta
+      columns 6
+      p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["4"]
+      i1["-"] i2["C"] i3["T"] i4["C"] i5["-"] i6["G"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef skip fill:#ffffff
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class s1,s5,i1,i5 skip
+      class s2,s4,i2,i4 c1
+      class s3,i3 c5
+      class s6,i6 c7
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+---
+
+$Bindings_p$, $Intervals_p$ and its inverse functions could be produced from functions defined for regular [_Bindings_, _Intervals chains_](../../order/intervals_chain/index.md#mathematical-definition) by adding condition that returns `-` for input `-` elements.
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier set_](../../order/carrier_set.md#mathematical-definition)
+
+Let $X_{-}$ is [_Partial Carrier set_](../carrier_set.md#mathematical-definition)
+
+Let $-$ is [_Empty element_](../carrier_set.md#mathematical-definition)
+
+Let $S$ is [_Sequence_](../../order/sequence.md#mathematical-definition) length of $l$ described as function $S : \{1,...,l\} \longrightarrow X$
+
+Let $S_p$ is [_Partial Sequence_](../sequence/index.md#mathematical-definition) length of $l$ described as function $S_p : \{1,...,l\} \longrightarrow X_{-}$
+
+Let $IC = <\Delta_1, \Delta_2, ..., \Delta_l> | \forall j \in \{1,...,l\} \exists \Delta_j \in \{1,...,l\}$ is [_Interval chain_](../../order/intervals_chain/index.md#define-intervals-chain)
+
+Let $Binding = <Iterator, \bot>$ is [_Binding_](../../order/intervals_chain/index.md#define-bindings)
+
+Let $R : \{1,...,l\} \longrightarrow \{1,...,l\} \cup \bot,$ is a corresponding [_references_](../../order/intervals_chain/index.md#define-bindings)
+
+Let $Intervals$ is [_Intervals function_](../../order/intervals_chain/index.md#define-intervals-chain) described as function
+
+$$Intervals : \big\{Binding\big\} \times \big\{S\}  \longrightarrow \big\{ IC \big\}$$
+
+Then _Iterator_ defined as $Iterator \big\{  S \big\} \longrightarrow \big\{ R \big\},$
+
+### Define Partial Bindings
+
+Let $R_p : \{1,...,l\} \longrightarrow \{1,...,l\} \cup \bot \cup \{-\},$ is a corresponding _partial references_
+
+Then
+
+$$X \subset X_{-},$$
+
+$$\{S\} \subset \{S_p\},$$
+
+$$\{R\} \subset \{R_p\},$$
+
+Define
+
+$$Iterator_p : \big\{  S_p \big\} \longrightarrow \big\{ R_p \big\},$$
+
+$$\{Iterator_p\} \supset \{Iterator\},$$
+
+$$Iterator_p(S_p)(i) = \Bigg \{ \begin{array}{l} Intertor(S_p)(i) , &  S_p(i) \notin \{-\} \\ -, & S_p(i) \in \{-\} \end{array},$$
+
+and
+
+$$Binding_p = <Iterator_p, \bot>,$$
+
+$$\{Binding_p\} \supset \{Binding\}$$
+
+the same way
+
+$$\exists Binding_p^{-1} = <Iterator_p^{-1}, \bot>,$$
+
+$$\{Binding_p^{-1}\} \supset \{Binding^{-1}\},$$
+
+where
+
+$$Iterator_p^{-1} : \big\{  R_p \big\} \longrightarrow \big\{  \{1,...,l\} \longrightarrow \{1,...,m\} \cup \{-\} | m \leq l \big\},$$
+
+$$\{Iterator_p^{-1}\} \supset \{Iterator^{-1}\},$$
+
+$$Iterator_p^{-1}(R_p)(i) = \Bigg \{ \begin{array}{l} Intertor^{-1}(R_p)(i) , &  R_p(i) \notin \{-\} \\ -, & R_p(i) \in \{-\} \end{array}$$
+
+The condition $Iterator_p(S_p) = Iterator_p(Iterator_p^{-1}(Iterator_p(S_p)))$ is valid.
+
+### Define Partial Intervals Chain
+
+as l-tuple of natural numbers and `-` empty elements
+
+$$IC_p = <\Delta_1, \Delta_2, ..., \Delta_l> | \forall j \in \{1,...,l\} \exists \Delta_j \in \{1,...,l\} \cup \{-\},$$
+
+Then
+
+$$\{IC_p\} \supset \{IC\},$$
+
+$$\exists \ Intervals_p : \big\{Binding_p\big\} \times \big\{S_p\}  \longrightarrow \big\{ IC_p \big\},$$
+
+$$\{Intervals_p\} \supset \{Intervals\},$$
+
+$$Intervals_p(b_p, S_p)(i) = \Bigg \{ \begin{array}{l} Intervals(b_p, S_p)(i) , &  S_p(i) \notin \{-\} \\ -, & S_p(i) \in \{-\} \end{array},$$
+
+$$\exists \ Intervals_p^{-1} : \big\{Binding_p^{-1}\big\} \times \big\{ IC_p \big\}  \longrightarrow \big\{ \{1,...,l\} \longrightarrow \{1,...,m\} \cup \{-\} | m \leq l \big\},$$
+
+$$\{Intervals_p^{-1}\} \supset \{Intervals^{-1}\},$$
+
+$$Intervals_p^{-1}(b_p^{-1}, IC_p)(i) = \Bigg \{ \begin{array}{l} Intervals^{-1}(b_p^{-1}, IC_p)(i) , &  IC_p(i) \notin \{-\} \\ -, & IC_p(i) \in \{-\} \end{array},$$
+
+The condition would be true
+
+$$Intervals_p(b_p, S_p) = Intervals_p(b_p, Intervals_p^{-1}(b_p^{-1}, Intervals_p(b_p, S_p)))$$
+
+Let $Follow$ is the [_follow function of interval chain_](../../order/intervals_chain/index.md#define-intervals-chain) described as function $Follow : \big\{Binding\big\} \times \big\{ IC \big\}  \longrightarrow \big\{ R \big\},$
+
+$$\exists \ Follow_p : \big\{Binding_p\big\} \times \big\{ IC_p \big\}  \longrightarrow \big\{ R_p \big\},$$
+
+$$\{Binding\} \subset \{Binding_p\},$$
+
+$$\{IC\} \subset \{IC_p\},$$
+
+$$\{R\} \subset \{R_p\},$$
+
+then
+
+$$\{Follow_p\} \supset \{Follow\}.$$
+
+$$Follow_p(b_p, IC_p)(i) = \Bigg \{ \begin{array}{l} Follow(b_p, IC_p)(i) , &  IC_p(i) \notin \{-\} \\ -, & IC_p(i) \in \{-\} \end{array},$$
+
+The condition would be true
+
+$$f = Follow_p(b_p, IC_p),$$
+
+$$f(i) <> f(j) \lor f(i) \in \bot | \forall i != j \land IC_p(i) \notin \{-\}.$$
+
+Let $Trace$ is the [_trace function of interval chain_](../../order/intervals_chain/index.md#define-intervals-chain) described as function $Trace : \big\{Binding\big\} \times \big\{ IC \big\} \longrightarrow \big\{ R \big\},$
+
+$$\exists \ Trace_p : \big\{Binding_p\big\} \times \big\{ IC_p \big\} \longrightarrow \big\{ R_p \big\},$$
+
+$$\{Binding\} \subset \{Binding_p\},$$
+
+$$\{IC\} \subset \{IC_p\},$$
+
+$$\{R\} \subset \{R_p\},$$
+
+then
+
+$$\{Trace_p\} \supset \{Trace\}.$$
+
+$$Trace_p(b_p, IC_p)(i) = \Bigg \{ \begin{array}{l} Trace_p(b_p, IC_p)(i) , &  IC_p(i) \notin \{-\} \\ -, & IC_p(i) \in \{-\} \end{array}.$$
+
+The condition would be true
+
+$$Trace_p(IC_p)(i) \in \bot | \forall i \in \{1,...,n\} \land IC_p(i) \notin \{-\}$$
+
+Where:
+
+- $l := |IC_p|$ is called _length_ of the _partial intervals chained_, $l \in N$
+- $n := |\{ IC_p(i) | IC_p(i) \ne - \}|$ is _non-empty elements count_, $n \in N$
+- $\Delta_i$​ is called the $i$-th _element_ (or interval) of the _partial intervals chained_
diff --git a/docs/fundamentals/partials_and_congenerics/intervals_distribution.md b/docs/fundamentals/partials_and_congenerics/intervals_distribution.md
new file mode 100644
index 00000000..94031a29
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/intervals_distribution.md
@@ -0,0 +1,110 @@
+# Partial Intervals Distribution
+
+A _Partial intervals distribution_ is an [_Interval distribution_](../order/intervals_distribution/index.md) produced from [_Partial intervals chain_](./intervals_chain/index.md) by counting all _non-empty_ elements (intervals) in distribution
+
+
+=== "From a partial interval chain"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+    ```
+
+    Let there be a _partial interval chain_.
+
+=== "calculate intervals distribution"
+
+    ``` mermaid
+    block-beta
+      columns 7
+      space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
+      space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]
+      space:7
+      space i1["0"] i2["2"] i3["1"] i4["0"] i5["0"] i6["1"]
+
+      classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
+      classDef position fill:#fff,color:#000,stroke-width:0px;
+      class inf,sup imaginary
+      class p0,p1,p2,p3,p4,p5,p6,p7 position
+
+      classDef c1 fill:#ff7f0e,color:#fff;
+      classDef c2 fill:#ffbb78,color:#000;
+      classDef c2a fill:#ffbb788a,color:#000;
+      classDef c3 fill:#2ca02c,color:#fff;
+      classDef c4 fill:#98df8a,color:#000;
+      classDef c4a fill:#98df8a8a,color:#000;
+      classDef c5 fill:#d62728,color:#fff;
+      classDef c6 fill:#ff9896,color:#000;
+      classDef c6a fill:#ff98968a,color:#000;
+      classDef c7 fill:#9467bd,color:#fff;
+      classDef c8 fill:#c5b0d5,color:#000;
+      classDef c9 fill:#8c564b,color:#fff;
+      classDef c10 fill:#c49c94,color:#000;
+      classDef c11 fill:#e377c2,color:#fff;
+      classDef c12 fill:#f7b6d2,color:#000;
+      classDef c13 fill:#bcbd22,color:#fff;
+      classDef c14 fill:#dbdb8d,color:#000;
+      classDef c14a fill:#dbdb8d8a,color:#000;
+      classDef c15 fill:#17becf,color:#fff;
+      classDef c16 fill:#9edae5,color:#000;
+
+      class pomn,p00,p01,p06,p07,p02n position
+      class t1,t2,t5,e0,e1 position
+
+      s2 --> i2
+      s3 --> i3
+      s4 --> i2
+      s6 --> i6
+
+
+    ```
+
+---
+
+_Partial Intervals distribution_, as any _Intervals distribution_, used as an input data in calculating [characteristics](../order/characteristics/index.md).
+
+## Mathematical Definition
+
+Let $-$ is [_empty element_](./carrier_set.md#mathematical-definition).
+
+Let $IC_p$ is [_Partial Interval Chain_](./intervals_chain/index.md#define-intervals-chain) length of $l$ described as function $IC_p : \{1,...,l\} \longrightarrow  \{1,...,l\} \cup \{-\}$
+
+Let $ID$ is [_Interval Distribution_](../order/intervals_distribution/index.md) length of $l$ described as function $ID : \big\{  IC \big\} \longrightarrow \big\{ \{1,...,l\} \longrightarrow  N_0 \big\},$
+
+Define
+
+$$ID_p : \big\{  IC_p \big\} \longrightarrow \big\{ \{1,...,l\} \longrightarrow  N_0 \big\},$$
+
+$$ID_p(IC_p)(i) = ID(IC_p)(i) \bigg|  IC_p(i) \notin \{-\}$$
diff --git a/docs/fundamentals/partials_and_congenerics/order/congeneric.md b/docs/fundamentals/partials_and_congenerics/order/congeneric.md
new file mode 100644
index 00000000..cd808b87
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/order/congeneric.md
@@ -0,0 +1,60 @@
+# Congeneric Order
+
+A _Congeneric order_ is an [_Partial Order_](./index.md) where all _non-empty_ elements equals $1$.
+
+
+``` mermaid
+block-beta
+columns 29
+
+i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+s1["-"] s2["-"] s3["1"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["1"] s18["-"] s19["-"] s20["-"]
+s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["1"] s27["-"] s28["-"] s29["1"]
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s1,s2,s4,s5,s6,s7,s8,s9,s10 skip
+class s11,s12,s13,s14,s15,s16,s18,s19,s20 skip
+class s21,s22,s23,s24,s25,s27,s28,s30 skip
+class s31,s32,s33,s34,s35 skip
+
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+```
+
+## Mathematical Definition
+
+Let $- \notin N$
+
+Let $N_{-} = N \cup \{-\}$
+
+Let $O_p$ is [_Partial order](index.md#mathematical-definition) $O_p : \{1, ..., n\} \longrightarrow N_{-},$
+
+$O_p$ is called $O_c$ _Congeneric order_ if
+
+$$\forall i O_{p}(i) \in \{-, 1\}$$
diff --git a/docs/fundamentals/partials_and_congenerics/order/index.md b/docs/fundamentals/partials_and_congenerics/order/index.md
new file mode 100644
index 00000000..fa591d58
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/order/index.md
@@ -0,0 +1,111 @@
+# Partial Order
+
+A _Partial order_ is an [_Order_](../../order/order.md) having _empty_ elements.
+
+
+``` mermaid
+block-beta
+columns 29
+
+i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+s1["1"] s2["2"] s3["3"] s4["4"] s5["5"] s6["5"] s7["1"] s8["6"] s9["4"] s10["2"]
+s11["7"] s12["4"] s13["8"] s14["-"] s15["-"] s16["8"] s17["3"] s18["-"] s19["-"] s20["8"]
+s21["-"] s22["9"] s23["-"] s24["5"] s25["-"] s26["3"] s27["-"] s28["-"] s29["3"]
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s14,s15,s18,s19,s21,s23,s25,s27,s28 skip
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+```
+
+## Mathematical Definition
+
+Let $-$ is [_Empty element_](../carrier_set.md#mathematical-definition)
+
+Let $- \notin N$
+
+Define $N_{-} = N \cup \{-\}$
+
+The _Partial order_ $O_p$ is defined as an l-tuple with additional constraints:
+
+$$O_p = <o_1, o_2, ..., o_l>,$$
+
+$$\forall i \in \{1, ..., l\} \exists o_i \in N_{-} $$
+
+$$ \exists j \in \{1, ..., l\}, \forall i < j | O_p(i) \in \{ -\} \land O_p(j) = 1  $$
+
+$$\forall i \in \{1, ..., l\}, O_p(i) \in \{-\} \lor O_p(i) \leq max(o_1, ..., o_{i-1}) + 1 \big| max(\{-\})=0$$
+
+Where:
+
+- $o_i$​ is called the $i$-th _element_ (or coordinate) of the partial order
+- $l := |O_p|$ is called _length_ of the partial order, $l \in N$
+- $n := |\{ O_p(i) | O_p(i) \ne - \}|$ is _non-empty elements count_, $n \in N$
+
+### Order of Partial Sequence
+
+Let $X$ is [_Carrier Set_](../../order/carrier_set.md#mathematical-definition)
+
+Let $X_{-}$ is [_Partial Carrier Set_](../carrier_set.md#mathematical-definition)
+
+$$X \subset X_{-}$$
+
+Let $-$ is [_Empty element_](../carrier_set.md#mathematical-definition) of _Partial Carrier Set_
+
+Let $S_p$ is [_Partial Sequence_](../sequence/index.md#mathematical-definition) described as function  $S_p : \{1,...,l\} \longrightarrow X_{-}$
+
+Let $alphabet_p$ is [_Alphabet of Partial Sequence function_](../alphabet/index.md#mathematical-definition)
+
+$$alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}$$
+
+Define
+
+$$order_p(S_p) :  \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,l\} \longrightarrow \{1,...,l\} \big\},$$
+
+$$A = alphabet_p(S_p),$$
+
+$$order_p(S_p)(i) = \Bigg\{\begin{array}{l} j \ \big| j \in \{1,...,l\}, S_p(i)=A(j), &  S_p(i) \notin \{-\} \\ -, & S_p(i) \in \{-\} \end{array}$$
+
+
+### Order product Alphabet
+
+Let $X$ is a [_Carrier set_](../../order/carrier_set.md)
+
+Let $A$ is a [_Aphabet_](../../order/alphabet.md) $A : \{1, ..., m\} \longrightarrow X,$
+
+Let $S_p$ is a [_Partial sequenece_](../sequence/index.md) $S : \{1, ..., l\} \longrightarrow X_{-},$
+
+the following equations are true
+
+$$O_p = order_p(S_p),$$
+
+$$A = alphabet_p(S_p),$$
+
+$$S_p = ( O_p \odot A),$$
+
+$$S_p(i) = \Bigg\{\begin{array}{l} A(O_p(i)) , &  O_p(i) \notin \{-\} \\ -, & O_p(i) \in \{-\} \end{array}$$
diff --git a/docs/fundamentals/partials_and_congenerics/sequence/congeneric.md b/docs/fundamentals/partials_and_congenerics/sequence/congeneric.md
new file mode 100644
index 00000000..5b15982f
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/sequence/congeneric.md
@@ -0,0 +1,60 @@
+# Congeneric Sequence
+
+_Congeneric sequence_ is an extreme case of [_Parial sequence_](./index.md) where all _non-empty_ elements are equals.
+
+ $S_{c|T}$ is _Congeneric sequence_ for `T`
+
+``` mermaid
+block-beta
+columns 29
+
+i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+s1["-"] s2["-"] s3["T"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+s11["-"] s12["-"] s13["-"] s14["-"] s15["-"] s16["-"] s17["T"] s18["-"] s19["-"] s20["-"]
+s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["T"] s27["-"] s28["-"] s29["T"]
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s1,s2,s4,s5,s6,s7,s8,s9,s10 skip
+class s11,s12,s13,s14,s15,s16,s18,s19,s20 skip
+class s21,s22,s23,s24,s25,s27,s28,s30 skip
+class s31,s32,s33,s34,s35 skip
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+```
+
+## Mathematical Definition
+
+Let $X$ is [_Carrier Set_](../../order/carrier_set.md#mathematical-definition)
+
+Let $X_{-}$ is [_Partial Carrier Set_](../carrier_set.md#mathematical-definition)
+
+$$X \subset X_{-}$$
+
+Let $S_p$ is [_Partial sequence_](index.md#mathematical-definition) $S_p : \{1, ..., l\} \longrightarrow X_{-},$
+
+$S_p$ is called $S_{c|e}$ - _Congeneric sequence_ of $e$ element - if $\exists e \in X\Big|\forall i, S_{p}(i) \in \{-, e\}$
diff --git a/docs/fundamentals/partials_and_congenerics/sequence/index.md b/docs/fundamentals/partials_and_congenerics/sequence/index.md
new file mode 100644
index 00000000..ce53f9ac
--- /dev/null
+++ b/docs/fundamentals/partials_and_congenerics/sequence/index.md
@@ -0,0 +1,171 @@
+# Partial Sequence
+
+_Partial sequence_ is a [_sequence_](../../order/sequence.md) where some elements are skipped.
+The concept can be treated as equivalent to the _masked sequence_ that is used in bioinformatics and data science.
+In FOA `-` symbol used as `empty` element.
+
+``` mermaid
+block-beta
+columns 29
+
+i1["1"] i2["2"] i3["3"] i4["4"] i5["5"] i6["6"] i7["7"] i8["8"] i9["9"] i10["10"] i11["11"] i12["12"]
+i13["13"] i14["14"] i15["15"] i16["16"] i17["17"] i18["18"] i19["19"] i20["20"]
+i21["21"] i22["22"] i23["23"] i24["24"] i25["25"] i26["26"] i27["27"]
+i28["28"] i29["29"]
+
+s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+s11["C"] s12["E"] s13[" "] s14["-"] s15["-"] s16[" "] s17["T"] s18["-"] s19["-"] s20[" "]
+s21["-"] s22["B"] s23["-"] s24["L"] s25["-"] s26["T"] s27["-"] s28["-"] s29["T"]
+
+classDef c1 fill:#ff7f0e,color:#fff;
+classDef c2 fill:#ffbb78,color:#000;
+classDef c3 fill:#2ca02c,color:#fff;
+classDef c4 fill:#98df8a,color:#000;
+classDef c5 fill:#d62728,color:#fff;
+classDef c6 fill:#ff9896,color:#000;
+classDef c7 fill:#9467bd,color:#fff;
+classDef c8 fill:#c5b0d5,color:#000;
+classDef c9 fill:#8c564b,color:#fff;
+classDef c10 fill:#c49c94,color:#000;
+classDef c11 fill:#e377c2,color:#fff;
+classDef c12 fill:#f7b6d2,color:#000;
+classDef c13 fill:#bcbd22,color:#fff;
+classDef c14 fill:#dbdb8d,color:#000;
+classDef c15 fill:#17becf,color:#fff;
+classDef c16 fill:#9edae5,color:#000;
+
+classDef skip fill:#ffffff
+classDef index fill:#ffffff,stroke-width:0px
+
+class s14,s15,s18,s19,s21,s23,s25,s27,s28 skip
+
+class i1,i2,i3,i4,i5,i6,i7,i8,i9,i10 index
+class i11,i12,i13,i14,i15,i16,i17,i18,i19,i20 index
+class i21,i22,i23,i24,i25,i26,i27,i28,i29 index
+```
+
+## Mathematical Definition
+
+Let $X_{-}$ is [_Partial Carrier set_](../carrier_set.md#mathematical-definition)
+
+A _Partial sequence_ $S_{p}$ is a l-tuple defined as
+
+$$S_{p} = <s_1, s_2, ..., s_l>,$$
+
+$$\forall i \in \{1, ..., l\} \exists s_i \in X_{-}$$
+
+where:
+
+- $s_i$​ is called the $i$-th _element_ (or coordinate) of the sequence.
+- $l := |S_p|$ is _length_, $l \in N$
+- $n := |\{ S_p(i) | S_p(i) \ne - \}|$ is _non-empty elements count_, $n \in N$
+
+The _sequence_ $S$ can be also defined as a function
+
+$$S_p : \{1, ..., l\} \longrightarrow X_{-},$$
+
+$$S_p(i)=s_i | i \in \{1, ..., l\}$$
+
+
+### Compatibility
+
+Two _Partial Sequences_ are called compatible if there are no _non-empty_ elements in the same position in both sequences.
+
+=== "Сompatible"
+
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+    s11["C"] s12["E"] s13[" "] s14["-"] s15["-"] s16[" "] s17["T"] s18["-"] s19["-"] s20[" "]
+    s21["-"] s22["B"] s23["-"] s24["L"] s25["-"] s26["T"] s27["-"] s28["-"] s29["T"] s30["-"]
+    s31["-"] s32["-"] s33["D"] s34["-"] s35["P"] s36["T"]
+
+    q1["-"] q2["-"] q3["-"] q4["-"] q5["-"] q6["-"] q7["-"] q8["-"] q9["-"] q10["-"]
+    q11["-"] q12["-"] q13["-"] q14["I"] q15["S"] q16["-"] q17["-"] q18["H"] q19["E"] q20["-"]
+    q21["A"] q22["-"] q23["I"] q24["-"] q25["I"] q26["-"] q27["I"] q28[" "] q29["-"] q30["O"]
+    q31[" "] q32["A"] q33["-"] q34["A"] q35["-"] q36["-"]
+
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#ffffff
+
+    class s14,s15,s18,s19,s21,s23,s25,s27,s28,s30,s31,s32,s34 skip
+
+    class q1,q2,q3,q4,q5,q6,q7,q8,q9,q10 skip
+    class q11,q12,q13,q16,q17,q20 skip
+    class q22,q24,q26,q29 skip
+    class q33,q35,q36 skip
+
+    ```
+
+=== "Incompatible"
+
+    ``` mermaid
+    block-beta
+    columns 36
+    s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"]
+    s11["C"] s12["E"] s13[" "] s14["-"] s15["-"] s16[" "] s17["T"] s18["-"] s19["-"] s20[" "]
+    s21["-"] s22["B"] s23["-"] s24["L"] s25["-"] s26["T"] s27["-"] s28["-"] s29["T"] s30["-"]
+    s31["-"] s32["-"] s33["D"] s34["-"] s35["P"] s36["T"]
+
+    q1["I"] q2["-"] q3["-"] q4["-"] q5["-"] q6["-"] q7["I"] q8["-"] q9["-"] q10["-"]
+    q11["-"] q12["-"] q13[" "] q14["I"] q15["S"] q16["-"] q17["-"] q18["H"] q19["E"] q20["-"]
+    q21["A"] q22["-"] q23["I"] q24["-"] q25["I"] q26["-"] q27["I"] q28[" "] q29["-"] q30["O"]
+    q31[" "] q32["A"] q33["-"] q34["A"] q35["-"] q36["-"]
+
+
+    classDef c1 fill:#ff7f0e,color:#fff;
+    classDef c2 fill:#ffbb78,color:#000;
+    classDef c3 fill:#2ca02c,color:#fff;
+    classDef c4 fill:#98df8a,color:#000;
+    classDef c5 fill:#d62728,color:#fff;
+    classDef c6 fill:#ff9896,color:#000;
+    classDef c7 fill:#9467bd,color:#fff;
+    classDef c8 fill:#c5b0d5,color:#000;
+    classDef c9 fill:#8c564b,color:#fff;
+    classDef c10 fill:#c49c94,color:#000;
+    classDef c11 fill:#e377c2,color:#fff;
+    classDef c12 fill:#f7b6d2,color:#000;
+    classDef c13 fill:#bcbd22,color:#fff;
+    classDef c14 fill:#dbdb8d,color:#000;
+    classDef c15 fill:#17becf,color:#fff;
+    classDef c16 fill:#9edae5,color:#000;
+
+    classDef skip fill:#ffffff
+
+    classDef conflict fill:#ff0000,color:#fff
+
+    class s14,s15,s18,s19,s21,s23,s25,s27,s28,s30,s31,s32,s34 skip
+
+    class q2,q3,q4,q5,q6,q8,q9,q10 skip
+    class q11,q12,q13,q16,q17,q20 skip
+    class q22,q24,q26,q29 skip
+    class q33,q35,q36 skip
+
+    class s1,q1,s7,q7 conflict
+    ```
+
+
+Let $S1_p$ and $S2_p$ are _Partial sequences_
+
+Define
+
+
+$$compatible(S1_p, S2_p) = \forall i \in \{1,...,l\}\ S1_p(i) \notin \{-\} \lor S2_p(i) \notin \{-\} $$
diff --git a/docs/references/characteristics/index.md b/docs/references/characteristics/index.md
index 347e0abd..a5afdbeb 100644
--- a/docs/references/characteristics/index.md
+++ b/docs/references/characteristics/index.md
@@ -9,21 +9,21 @@ The package provides a comprehensive set of characteristics for measuring the pr
 
 The table below summarizes the available characteristics that depend only on intervals:
 
-| Linear scale | || Logarifmic scale | |
+| Linear scale | || Logarithmic scale | |
 |------------- |-||-|-----------------|
-| [Arithmetic Mean](/references/characteristics/arithmetic_mean/) | $\Delta_a = \frac{1}{n} * \sum_{i=1}^{n} \Delta_{i}$ || | |
-| [Geometric Mean](/references/characteristics/geometric_mean/) | $\Delta_g=\sqrt[n]{\prod_{i=1}^{n} \Delta_{i}}$ || $g = \frac{1}{n} * \sum_{i=1}^{n} \log_2 \Delta_{i}$ | [Average Remoteness](/references/characteristics/average_remoteness/) |
-| [Volume](/references/characteristics/volume/) | $V=\prod_{i=1}^{n} \Delta_{i}$ || $G=\sum_{i=1}^{n} \log_2 \Delta_{i}$ | [Depth](/references/characteristics/depth/) |
+| [Arithmetic Mean](arithmetic_mean.md) | $\Delta_a = \frac{1}{n} * \sum_{i=1}^{n} \Delta_{i}$ || | |
+| [Geometric Mean](geometric_mean.md) | $\Delta_g=\sqrt[n]{\prod_{i=1}^{n} \Delta_{i}}$ || $g = \frac{1}{n} * \sum_{i=1}^{n} \log_2 \Delta_{i}$ | [Average Remoteness](average_remoteness.md) |
+| [Volume](volume.md) | $V=\prod_{i=1}^{n} \Delta_{i}$ || $G=\sum_{i=1}^{n} \log_2 \Delta_{i}$ | [Depth](depth.md) |
 
 
 The table below summarizes the available characteristics that depend on cogeneric intervals ( grouped by element of the alphabet):
 
 | Characteristics   |                                                                                      |
 |-------------------------------|---------------------------------------------------------------------------------------------------------|
-| [Descriptive Information](/references/characteristics/descriptive_information/) | $D=\prod_{j=1}^{m}{\left(\sum_{i=1}^{n_j}{\frac{\Delta_{ij}}{n_j}}\right)^{\frac{n_j}{n}}}$                          |
-| [Identifying Information](/references/characteristics/identifying_information/) | $H=\frac {1} {n} * \sum_{j=1}^{m}{(n_j * \log_2 \sum_{i=1}^{n_j} \frac{\Delta_{ij}}{n_j})}$ |
-| [Regularity](/references/characteristics/regularity/)                     | $r= \sqrt[n]{\prod_{j=1}^{m} \frac{\prod_{j=1}^{n_j} \Delta_{ij}}{{\left(\frac{1}{n_j}\sum_{i=1}^{n_j}{\Delta_{ij}}\right)^{n_j}}}}$  |
-| [Uniformity](/references/characteristics/uniformity/)                     | $u = \frac {1} {n} * \sum_{j=1}^{m}{\log_2 \frac{ (\sum_{i=1}^{n_j} \frac{\Delta_{ij}}{n_j})^{n_j} } { \prod_{i=1}^{n_j} \Delta_{ij}}}$                            |
+| [Descriptive Information](descriptive_information.md) | $D=\prod_{j=1}^{m}{\left(\sum_{i=1}^{n_j}{\frac{\Delta_{ij}}{n_j}}\right)^{\frac{n_j}{n}}}$                          |
+| [Identifying Information](identifying_information.md) | $H=\frac {1} {n} * \sum_{j=1}^{m}{(n_j * \log_2 \sum_{i=1}^{n_j} \frac{\Delta_{ij}}{n_j})}$ |
+| [Regularity](regularity.md)                     | $r= \sqrt[n]{\prod_{j=1}^{m} \frac{\prod_{j=1}^{n_j} \Delta_{ij}}{{\left(\frac{1}{n_j}\sum_{i=1}^{n_j}{\Delta_{ij}}\right)^{n_j}}}}$  |
+| [Uniformity](uniformity.md)                     | $u = \frac {1} {n} * \sum_{j=1}^{m}{\log_2 \frac{ (\sum_{i=1}^{n_j} \frac{\Delta_{ij}}{n_j})^{n_j} } { \prod_{i=1}^{n_j} \Delta_{ij}}}$                            |
 
 
-[ma](/references/characteristics/ma/) subpackage provides characteristics for cogeneric intervals ( grouped by element).
+[ma](ma/index.md) subpackage provides characteristics for cogeneric intervals ( grouped by element).
diff --git a/docs/references/characteristics/ma/index.md b/docs/references/characteristics/ma/index.md
index 8d84d7e4..4f168a67 100644
--- a/docs/references/characteristics/ma/index.md
+++ b/docs/references/characteristics/ma/index.md
@@ -8,17 +8,17 @@ The package provides a comprehensive set of vector characteristics for measuring
 
 The table below summarizes vector representation of the characteristics that depend only on intervals:
 
-| Linear scale | |Logarifmic scale | |
+| Linear scale | |Logarithmic scale | |
 |------------- |-||-----------------|
-| [Arithmetic Mean](/references/characteristics/ma/arithmetic_mean/) | $\left[ \Delta_{a_j} \right]_{1 \le j \le m} = \left[ \frac{1}{n_j} * \sum_{i=1}^{n_j} \Delta_{ij} \right]_{1 \le j \le m}$ || |
-| [Geometric Mean](/references/characteristics/ma/geometric_mean/) | $\left[ \Delta_{g_j} \right]_{1 \le j \le m} = \left[ \left( \prod_{i=1}^{n_j} \Delta_{ij} \right)^{1/n_j} \right]_{1 \le j \le m}$ | $\left[ g_j \right]_{1 \le j \le m} = \left[ \frac{1}{n_j} * \sum_{i=1}^{n_j} \log_2 \Delta_{ij} \right]_{1 \le j \le m}$ | [Average Remoteness](/references/characteristics/ma/average_remoteness/) |
-| [Volume](/references/characteristics/ma/volume/) | $\left[ V_j \right]_{1 \le j \le m} = \left[ \prod_{i=1}^{n_j} \Delta_{ij} \right]_{1 \le j \le m}$  |$\left[ G_j \right]_{1 \le j \le m} = \left[  \sum_{i=1}^{n_j} \log_2 \Delta_{ij} \right]_{1 \le j \le m}$|  [Depth](/references/characteristics/ma/depth/) |
+| [Arithmetic Mean](arithmetic_mean.md) | $\left[ \Delta_{a_j} \right]_{1 \le j \le m} = \left[ \frac{1}{n_j} * \sum_{i=1}^{n_j} \Delta_{ij} \right]_{1 \le j \le m}$ || |
+| [Geometric Mean](geometric_mean.md) | $\left[ \Delta_{g_j} \right]_{1 \le j \le m} = \left[ \left( \prod_{i=1}^{n_j} \Delta_{ij} \right)^{1/n_j} \right]_{1 \le j \le m}$ | $\left[ g_j \right]_{1 \le j \le m} = \left[ \frac{1}{n_j} * \sum_{i=1}^{n_j} \log_2 \Delta_{ij} \right]_{1 \le j \le m}$ | [Average Remoteness](average_remoteness.md) |
+| [Volume](volume.md) | $\left[ V_j \right]_{1 \le j \le m} = \left[ \prod_{i=1}^{n_j} \Delta_{ij} \right]_{1 \le j \le m}$  |$\left[ G_j \right]_{1 \le j \le m} = \left[  \sum_{i=1}^{n_j} \log_2 \Delta_{ij} \right]_{1 \le j \le m}$|  [Depth](depth.md) |
 
 
 The table below summarizes the advanced characteristics of cogeneric intervals:
 
 | Characteristics   |                                                                                      |
 |-------------------------------|---------------------------------------------------------------------------------------------------------|
-| [Identifying Information](/references/characteristics/ma/identifying_information/) | $\left[ H_j \right]_{1 \le j \le m} = \left[ \log_2 { \left(\frac{1}{n_j} * \sum_{i=1}^{n_j} \Delta_{ij} \right) } \right]_{1 \le j \le m}$ |
-| [Periodicity](/references/characteristics/ma/periodicity/)                     | $\left[ \tau_j \right]_{1 \le j \le m} = \left[ \left( \prod_{i=1}^{n_j} \Delta_{ij} \right)^{1/n_j} * \frac{ n_j }{ \sum_{i=1}^{n_j} \Delta_{ij} } \right]_{1 \le j \le m}$  |
-| [Uniformity](/references/characteristics/ma/uniformity/)                     | $\left[ u_j \right]_{1 \le j \le m} = \left[ \log_2 { \left(\frac{1}{n_j} * \sum_{i=1}^{n_j} \Delta_{ij} \right) } - \frac{1}{n_j} * \sum_{i=1}^{n_j} \log_2 \Delta_{ij} \right]_{1 \le j \le m}$                            |
+| [Identifying Information](identifying_information.md) | $\left[ H_j \right]_{1 \le j \le m} = \left[ \log_2 { \left(\frac{1}{n_j} * \sum_{i=1}^{n_j} \Delta_{ij} \right) } \right]_{1 \le j \le m}$ |
+| [Periodicity](periodicity.md)                     | $\left[ \tau_j \right]_{1 \le j \le m} = \left[ \left( \prod_{i=1}^{n_j} \Delta_{ij} \right)^{1/n_j} * \frac{ n_j }{ \sum_{i=1}^{n_j} \Delta_{ij} } \right]_{1 \le j \le m}$  |
+| [Uniformity](uniformity.md)                     | $\left[ u_j \right]_{1 \le j \le m} = \left[ \log_2 { \left(\frac{1}{n_j} * \sum_{i=1}^{n_j} \Delta_{ij} \right) } - \frac{1}{n_j} * \sum_{i=1}^{n_j} \log_2 \Delta_{ij} \right]_{1 \le j \le m}$                            |
diff --git a/docs/references/exceptions/index.md b/docs/references/exceptions/index.md
index 7007095f..f527da66 100644
--- a/docs/references/exceptions/index.md
+++ b/docs/references/exceptions/index.md
@@ -10,5 +10,5 @@ The table below summarizes the available exceptions:
 
 | Exception | Description |
 |-----------|-------------|
-| [Not1DArrayException](/references/exceptions/not_1d_array/) | Raised when function receives not 1D array as input. |
-| [InconsistentOrderException](/references/exceptions/inconsistent_order/) | Raised when function receives inconsistent order. |
+| [Not1DArrayException](not_1d_array.md) | Raised when function receives not 1D array as input. |
+| [InconsistentOrderException](inconsistent_order.md) | Raised when function receives inconsistent order. |
diff --git a/docs/references/ma/index.md b/docs/references/ma/index.md
index 82f88ea7..4eef273a 100644
--- a/docs/references/ma/index.md
+++ b/docs/references/ma/index.md
@@ -6,4 +6,4 @@ hide:
 
 The package provides a comprehensive set of functions that decompose sequence into cogeneric order and congeneric intervals.
 
-The functions required by [vector representation of characteristics](/references/characteristics/ma) and [characteristics of order that depends on congeneric intervals distribution](/references/characteristics) are:
+The functions required by vector representation of characteristics and [characteristics of order that depends on congeneric intervals distribution](../characteristics/index.md) are:
diff --git a/mkdocs.yml b/mkdocs.yml
index 8975d29b..6608f606 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -6,50 +6,74 @@ nav:
   - fundamentals/index.md
   - "Ideas":
     - fundamentals/ideas/index.md
-    - "Sequence as a whole object": fundamentals/ideas/sequence_as_an_system_object.md
-    - "Order as a sequence property": fundamentals/ideas/order_as_a_sequence_property.md
+    - "Sequence as a whole object": fundamentals/ideas/sequence_as_a_whole_object.md
+    - "Order as a property": fundamentals/ideas/order_as_a_property.md
     - "Congeneric decomposition": fundamentals/ideas/congeneric_decomposition.md
     - "Interval as a basic information unit": fundamentals/ideas/interval_as_a_basic_information_unit.md
     - "Geomteric mean as alternative to probability": fundamentals/ideas/geometric_mean_based_characteristics.md
-  - "Objects":
-    - fundamentals/objects/index.md
-    - "Sequence": fundamentals/objects/sequence.md
-    - "Alphabet": fundamentals/objects/alphabet.md
-    - "Order": fundamentals/objects/order.md
-    - "Interval": fundamentals/objects/interval.md
-    - "Intervals Order": fundamentals/objects/intervals_order.md
-    - "Intervals Tuple": fundamentals/objects/intervals_tuple.md
-    - "Intervals Distribution": fundamentals/objects/intervals_distribution.md
+  - "Order and its measures":
+    - fundamentals/order/index.md
+    - "Carrier set": fundamentals/order/carrier_set.md
+    - "Sequence": fundamentals/order/sequence.md
+    - "Alphabet": fundamentals/order/alphabet.md
+    - "Order": fundamentals/order/order.md
+    - "Intervals Chain":
+      - fundamentals/order/intervals_chain/index.md
+      - "Bounded": fundamentals/order/intervals_chain/bounded.md
+      - "Cycled": fundamentals/order/intervals_chain/cycled.md
+    - "Intervals Distribution":
+      - fundamentals/order/intervals_distribution/index.md
+      - "Lossy": fundamentals/order/intervals_distribution/lossy.md
+      - "Redundant": fundamentals/order/intervals_distribution/redundant.md
+    - "Characteristics":
+      - fundamentals/order/characteristics/index.md
+      - "Arithmetic mean": fundamentals/order/characteristics/arithmetic_mean.md
+      - "Geometric mean": fundamentals/order/characteristics/geometric_mean.md
+      - "Volume": fundamentals/order/characteristics/volume.md
+      - "Average remoteness": fundamentals/order/characteristics/average_remoteness.md
+      - "Depth": fundamentals/order/characteristics/depth.md
+  - "Partials and congenerics":
+    - fundamentals/partials_and_congenerics/index.md
+    - "Carrier set": fundamentals/partials_and_congenerics/carrier_set.md
+    - "Sequence":
+      - fundamentals/partials_and_congenerics/sequence/index.md
+      - "Congeneric": fundamentals/partials_and_congenerics/sequence/congeneric.md
+    - "Alphabet":
+      - fundamentals/partials_and_congenerics/alphabet/index.md
+      - "Congeneric": fundamentals/partials_and_congenerics/alphabet/congeneric.md
+    - "Order":
+      - fundamentals/partials_and_congenerics/order/index.md
+      - "Congeneric": fundamentals/partials_and_congenerics/order/congeneric.md
+    - "Intervals chain":
+      - fundamentals/partials_and_congenerics/intervals_chain/index.md
+      - "Congeneric": fundamentals/partials_and_congenerics/intervals_chain/congeneric.md
+    - "Intervals distribution": fundamentals/partials_and_congenerics/intervals_distribution.md
+    - "Characteristics": fundamentals/partials_and_congenerics/characteristics.md
   - "Congeneric decomposition":
     - fundamentals/congeneric_decomposition/index.md
-    - "Congeneric sequences": fundamentals/congeneric_decomposition/congeneric_sequences.md
-    - "Congeneric orders": fundamentals/congeneric_decomposition/congeneric_orders.md
-    - "Congeneric intervals order": fundamentals/congeneric_decomposition/congeneric_intervals_order.md
-    - "Congeneric intervals tuple": fundamentals/congeneric_decomposition/congeneric_intervals_tuple.md
-    - "Congeneric intervals distribution": fundamentals/congeneric_decomposition/congeneric_intervals_distribution.md
-  - "Characteristics":
-    - fundamentals/characteristics/index.md
-    - "Arithmetic mean": fundamentals/characteristics/arithmetic_mean.md
-    - "Average remoteness": fundamentals/characteristics/average_remoteness.md
-    - "Depth": fundamentals/characteristics/depth.md
-    - "Descriptive information": fundamentals/characteristics/descriptive_information.md
-    - "Geometric mean": fundamentals/characteristics/geometric_mean.md
-    - "Identifying information": fundamentals/characteristics/identifying_information.md
-    - "Periodicity": fundamentals/characteristics/periodicity.md
-    - "Uniformity": fundamentals/characteristics/uniformity.md
-    - "Volume": fundamentals/characteristics/volume.md
-  - "Connections to other mathfields":
-    - fundamentals/connections_to_other_mathfields/index.md
-    - "Information theory": fundamentals/connections_to_other_mathfields/information_theory.md
-    - "Probability theory": fundamentals/connections_to_other_mathfields/probability_theory.md
-    - "Statistics": fundamentals/connections_to_other_mathfields/statistics.md
-    - "Combinatorics": fundamentals/connections_to_other_mathfields/combinatorics.md
-  - "Applications":
-    - fundamentals/applications/index.md
-    - "Text mining": fundamentals/applications/text_mining.md
-    - "Bioinformatics": fundamentals/applications/bioinformatics.md
-    - "Music analysis": fundamentals/applications/music_analysis.md
-    - "Other applications": fundamentals/applications/other_applications.md
+    - "Sequences": fundamentals/congeneric_decomposition/sequences.md
+    - "Alphabet": fundamentals/congeneric_decomposition/alphabet.md
+    - "Orders": fundamentals/congeneric_decomposition/orders.md
+    - "Intervals chains": fundamentals/congeneric_decomposition/intervals_chains.md
+    - "Intervals distribution": fundamentals/congeneric_decomposition/intervals_distribution.md
+    - "Characteristics":
+      - fundamentals/congeneric_decomposition/characteristics/index.md
+      - "Descriptive informations": fundamentals/congeneric_decomposition/characteristics/descriptive_information.md
+      - "Identifying information": fundamentals/congeneric_decomposition/characteristics/identifying_information.md
+      - "Regularity": fundamentals/congeneric_decomposition/characteristics/regularity.md
+      - "Uniformity": fundamentals/congeneric_decomposition/characteristics/uniformity.md
+  # - "Connections to other mathfields":
+  #   - fundamentals/connections_to_other_mathfields/index.md
+  #   - "Information theory": fundamentals/connections_to_other_mathfields/information_theory.md
+  #   - "Probability theory": fundamentals/connections_to_other_mathfields/probability_theory.md
+  #   - "Statistics": fundamentals/connections_to_other_mathfields/statistics.md
+  #   - "Combinatorics": fundamentals/connections_to_other_mathfields/combinatorics.md
+  # - "Applications":
+  #   - fundamentals/applications/index.md
+  #   - "Text mining": fundamentals/applications/text_mining.md
+  #   - "Bioinformatics": fundamentals/applications/bioinformatics.md
+  #   - "Music analysis": fundamentals/applications/music_analysis.md
+  #   - "Other applications": fundamentals/applications/other_applications.md
 - "References":
   - "foapy.alphabet": references/alphabet.md
   - "foapy.order": references/order.md
@@ -138,12 +162,16 @@ markdown_extensions:
       pygments_lang_class: true
   - pymdownx.inlinehilite
   - pymdownx.snippets
-  - pymdownx.superfences
   - tables
   - pymdownx.tabbed:
       alternate_style: true
   - pymdownx.arithmatex:
       generic: true
+  - pymdownx.superfences:
+      custom_fences:
+        - name: mermaid
+          class: mermaid
+          format: !!python/name:pymdownx.superfences.fence_code_format
 
 plugins:
 - autorefs
diff --git a/src/foapy/characteristics/_regularity.py b/src/foapy/characteristics/_regularity.py
index 17768cc3..b41a8ce0 100644
--- a/src/foapy/characteristics/_regularity.py
+++ b/src/foapy/characteristics/_regularity.py
@@ -1,7 +1,7 @@
 import numpy as np
 
 
-def regularity(intervals, dtype=None):
+def regularity(intervals_grouped, dtype=None):
     """
     Calculates regularity of intervals grouped by element of the alphabet.
 
@@ -75,7 +75,7 @@ def regularity(intervals, dtype=None):
     """  # noqa: E501
     from foapy.characteristics import descriptive_information, geometric_mean
 
-    total_elements = np.concatenate(intervals)
+    total_elements = np.concatenate(intervals_grouped)
     g = geometric_mean(total_elements, dtype=dtype)
-    D = descriptive_information(intervals, dtype=dtype)
+    D = descriptive_information(intervals_grouped, dtype=dtype)
     return g / D
diff --git a/src/foapy/characteristics/_uniformity.py b/src/foapy/characteristics/_uniformity.py
index a9078ae3..31e6eb8e 100644
--- a/src/foapy/characteristics/_uniformity.py
+++ b/src/foapy/characteristics/_uniformity.py
@@ -1,7 +1,7 @@
 import numpy as np
 
 
-def uniformity(intervals, dtype=None):
+def uniformity(intervals_grouped, dtype=None):
     """
     Calculates uniformity of intervals grouped by element of the alphabet.
 
@@ -71,9 +71,9 @@ def uniformity(intervals, dtype=None):
     """  # noqa: E501
     from foapy.characteristics import average_remoteness, identifying_information
 
-    total_elements = np.concatenate(intervals)
+    total_elements = np.concatenate(intervals_grouped)
 
-    H = identifying_information(intervals, dtype=dtype)
+    H = identifying_information(intervals_grouped, dtype=dtype)
     g = average_remoteness(total_elements, dtype=dtype)
 
     return H - g
diff --git a/src/foapy/ma/_intervals.py b/src/foapy/ma/_intervals.py
index 8c001ef0..26b652c8 100644
--- a/src/foapy/ma/_intervals.py
+++ b/src/foapy/ma/_intervals.py
@@ -1,11 +1,12 @@
 import numpy as np
 from numpy import ma
 
-from foapy import binding, mode
+from foapy import binding as binding_enum
+from foapy import mode as mode_enum
 from foapy.exceptions import InconsistentOrderException, Not1DArrayException
 
 
-def intervals(X, bind, mod):
+def intervals(X, binding, mode):
     """
     Finding array of array of intervals of the uniform
     sequences in the given input sequence
@@ -157,13 +158,19 @@ def intervals(X, bind, mod):
     """
 
     # Validate binding
-    if bind not in {binding.start, binding.end}:
+    if binding not in {binding_enum.start, binding_enum.end}:
         raise ValueError(
             {"message": "Invalid binding value. Use binding.start or binding.end."}
         )
 
     # Validate mode
-    if mod not in {mode.lossy, mode.normal, mode.cycle, mode.redundant}:
+    valid_modes = {
+        mode_enum.lossy,
+        mode_enum.normal,
+        mode_enum.cycle,
+        mode_enum.redundant,
+    }
+    if mode not in valid_modes:
         raise ValueError(
             {"message": "Invalid mode value. Use mode.lossy,normal,cycle or redundant."}
         )
@@ -199,7 +206,7 @@ def intervals(X, bind, mod):
         )
 
     extended_mask = np.empty((power, length + 1), dtype=bool)
-    if bind == binding.end:
+    if binding == binding_enum.end:
         extended_mask[:, :-1] = ~mask[::-1, ::-1]
     else:
         extended_mask[:, :-1] = ~mask
@@ -217,33 +224,33 @@ def intervals(X, bind, mod):
 
     indecies = np.zeros(positions.shape[1], dtype=int)
     indecies[1:] = positions[1, 1:] - positions[1, :-1]
-    delta = indecies[last_indexes] if mod == mode.cycle else 1
+    delta = indecies[last_indexes] if mode == mode_enum.cycle else 1
     indecies[first_indexes] = positions[1][first_indexes] + delta
 
     split_boarders = np.zeros(power * 2, dtype=int)
 
-    if mod == mode.lossy:
+    if mode == mode_enum.lossy:
         split_boarders[positions[0][last_indexes[:1]] * 2] = first_indexes[:1] + 1
         split_boarders[positions[0][last_indexes[1:]] * 2] = (
             np.fmax(last_indexes[:-1], first_indexes[1:]) + 1
         )
         split_boarders[positions[0][last_indexes] * 2 + 1] = last_indexes
-    elif mod == mode.normal:
+    elif mode == mode_enum.normal:
         split_boarders[positions[0][last_indexes[:1]] * 2] = 0
         split_boarders[positions[0][last_indexes[1:]] * 2] = last_indexes[:-1] + 1
         split_boarders[positions[0][last_indexes] * 2 + 1] = last_indexes
-    elif mod == mode.cycle:
+    elif mode == mode_enum.cycle:
         split_boarders[positions[0][last_indexes[:1]] * 2] = 0
         split_boarders[positions[0][last_indexes[1:]] * 2] = last_indexes[:-1] + 1
         split_boarders[positions[0][last_indexes] * 2 + 1] = last_indexes
-    elif mod == mode.redundant:
+    elif mode == mode_enum.redundant:
         split_boarders[positions[0][last_indexes[:1]] * 2] = 0
         split_boarders[positions[0][last_indexes[1:]] * 2] = 0
         split_boarders[positions[0][last_indexes] * 2 + 1] = last_indexes + 1
 
     preserve_previous = np.frompyfunc(lambda x, y: x if y == 0 else y, 2, 1)
     split_boarders = preserve_previous.accumulate(split_boarders)
-    if bind == binding.end:
+    if binding == binding_enum.end:
         split_boarders[:-1] = np.diff(split_boarders)
         split_boarders[-1:] = len(indecies) - split_boarders[-1]
         split_boarders = np.cumsum(split_boarders[::-1])