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User:Watchduck/hat

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<User:Watchduck

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

Terms

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The termhaploid is used as a synonym forself-complementary. Objects forming pairs of distinct complements arediploid.(The terms can be used for equivalence classes of Boolean functions, e.g.families.)

Reverse binary

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When finite subsets are to be ordered in a sequence, it is often better to order them like reversed binary numbers - although for most people ordering them like binary numbers would be more intuitive.
For some years I have called this "little-endian binary" — butendianness refers to the order of bytes (not of bits). The term does not make sense outside of computer science.

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

{			}
{		B	}
{	A		}
{	A,	B	}

The subsets of {A,B,C}
ordered like reversed 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 reversed 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

variadicAND of arguments in reverse binary order

The elements of $~V_{4}~$ ordered in a Hasse diagram
This enumeration is the only natural one for elements of $~V_{\omega }~$ .
(compare:Hereditarily finite set)

A more general concept iscolexicographic order (seelexicographic and colexicographic order).

Dual matrix

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16×16 matrix of 1×4 matrices

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

(unreversed)

4x16 matrix of 1x4 matrices

Below the dual 1x4 matrix
of 4x16 matrices

(reversed)

When a matrixA is anm×n matrix, containingp×q matricesB_ij as elements,
it is often interesting to see thedual matrixX, which is ap×q matrix, containingm×n matricesY_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 matrixB_ij
is the same as
the element y_kl,ij in the matrixY_kl .

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

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

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

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

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
Join table of the weak order of permutations This file shows the same bits like the join table above, but with a single matrix for each inversion. (Compare:Symmetric_group_S4#Join_and_meet)

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

Boolean functions

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For equivalence classes like families and clans seeProperties of Boolean functions.

Nibble shorthands

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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 thecomplementary nibble.
(Hence the vertical reflection gives the reflected complement, which is e.g. the relationship betweenAND andOR).
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 only the nibbles. There are two ways to assign them to integers, namely as binary or reverse binary numbers.
The similarity of and to 3 and 7 makes the unreversed interpretation more intuitive.(In that interpretation they arehexadecimal figures.)

Representingsubsets of a 4-element set. The connection tothis file suggests thereversed interpretation.

Representinglogical connectives. The representation of Boolean functions asrationals (rather than integers) suggests theunreversed interpretation. E.g. represents the fraction 3/15 = 1/5 with the period... after thepoint.

Representing numbers 0...15 with theunreversed interpretation.

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