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[edit]

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

Nibble shorthands[edit]

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[edit]

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[edit]

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

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


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} 
.

b_{11,11} = a = y_{11,11}~

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.