User:Watchduck/hat

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 does not exist, and have to choose one on my own.

For equivalence classes like SECs and BECs see Equivalence classes of Boolean functions.

commons:Category:Nibble shorthands

For some purposes I use a set of self-developed signs for the 16 binary strings with 4 digits (nibbles).

Their horizontal symmetry is like that of the nibbles themselves. The rotation of a sign belongs to the complementary nibble.
(Hence the vertical reflection gives the reflected complement, which is e.g. the relationship between AND and OR).
The signs of asymmetric nibbles have their weight on the same side as the nibble (i.e. the side with more 1s).

The signs represent the nibbles themselves, and not anything they stand for.
But the similarity of and to 3 and 7 makes the big-endian interpretation (as hexadecimal figures) more intuitive than little-endian.

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.

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...}.

examples

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

When a matrix A is an m×n matrix, containing p×q matrices B_ij as elements,
it is often interesting to see the dual matrix X, which is a p×q matrix, containing m×n matrices Y_ij 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 b_ij,kl in the matrix B_ij
is the same as
the element y_kl,ij in the matrix Y_kl .

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

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

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

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

The following example is a 24×24 join table containing triangular grids with binary entries, and below the corresponding triangular grid containing 24x24 matrices with binary entries.

Join table
(Compare: Symmetric_group_S4#Join_and_meet)

Vertex colored graphs can have the same kind of duality. See e.g. the tesseract graph above.