# User:Watchduck/hat

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Habits and terminology

I try to use technical terms that are generally accepted, but sometimes I don't know a common name,
possibly because it doesn't exist, and have to choose one on my own.

## Boolean functions

For equivalence classes like secs and becs see Equivalence classes of Boolean functions.

### Nibble shorthands

commons:Category:Nibble shorthands
Tesseract Hasse diagram
with nibble shorthands

For some purposes I use a set of self-developed symbols for the 16 binary strings with 4 digits (nibbles).
They just represent the strings themselves, and not anything they may stand for.

They may be assigned to numbers from 0 to 15 like binary or like reverse binary numbers, I usually do the latter.

## Reverse binary

Reverse binary warning sign

When finite subsets are to be ordered in a sequence, it is often better to order them like reflected binary numbers (little-endian) - although for most people ordering them like binary numbers would be more intuitive.

The subsets of {A,B}
ordered like binary numbers are:

 { } { B } { A } { A, B }
The subsets of {A,B,C}
ordered like reflected binary numbers are:
 { } { A } { B } { A, B } { C } { A, C } { B, C } { A, B, C }
The subsets of {A,B,C}
ordered like binary numbers are:
 { } { C } { B } { B, C } { A } { A, C } { A, B } { A, B, C }

Only when the subsets are ordered like reflected binary numbers, the sequence of subsets of {A,B}
is the beginning of the sequence of subsets of {A,B,C}.
This leads to a sequence of finite subsets of the infinite set {A,B,C,D...}.

A more general concept is colexicographic order (see lexicographic and colexicographic order).

## Dual matrix

16×16 matrix of 1×4 matrices

Below the dual 1×4 matrix
of 16×16 matrices
4x16 matrix of 1x4 matrices

Below the dual 1x4 matrix
of 4x16 matrices

When a matrix A is an m×n matrix, containing p×q matrices Bij as elements,
it is often interesting to see the dual matrix X, which is a p×q matrix, containing m×n matrices Yij as elements.

Dual matrices contain the same elements of elements (usually that should be numbers),
so in the end they show the same information, but in a different way.

The element bij,kl in the matrix Bij
is the same as
the element ykl,ij in the matrix Ykl .

The matrix
${\displaystyle A={\begin{pmatrix}B_{11}&B_{12}&B_{13}\\B_{21}&B_{22}&B_{23}\end{pmatrix}}={\begin{pmatrix}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}&{\begin{pmatrix}1&2\\3&4\end{pmatrix}}&{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}\\{\begin{pmatrix}e&f\\g&h\end{pmatrix}}&{\begin{pmatrix}5&6\\7&8\end{pmatrix}}&{\begin{pmatrix}\epsilon &\zeta \\\eta &\theta \end{pmatrix}}\end{pmatrix}}}$

is dual to

${\displaystyle X={\begin{pmatrix}Y_{11}&Y_{12}\\Y_{21}&Y_{22}\end{pmatrix}}={\begin{pmatrix}{\begin{pmatrix}a&1&\alpha \\e&5&\epsilon \end{pmatrix}}&{\begin{pmatrix}b&2&\beta \\f&6&\zeta \end{pmatrix}}\\{\begin{pmatrix}c&3&\gamma \\g&7&\eta \end{pmatrix}}&{\begin{pmatrix}d&4&\delta \\h&8&\theta \end{pmatrix}}\end{pmatrix}}}$ .

${\displaystyle b_{11,11}=a=y_{11,11}~}$

${\displaystyle b_{23,21}=\eta =y_{21,23}~}$

This concept is not limited to matrices, as the following example shows.
The join table is a 24×24 matrix. The inversion sets could be displayed by 6-bit binary vectors,
but a free arrangement was chosen, to serve symmetry.