Seal (discrete mathematics)

Studies of Boolean functions

Seal is a neologism for a mathematical object, that is essentially a subgroup of nimber addition.
The addition of nimbers is the bitwise XOR of non-negative integers. For a finite set it forms the Boolean group Z2n.

A seal shall be defined as a Boolean function whose family matrix is also the matrix of an equivalence relation.
This implies, that the Boolean function is odd (i.e. that the first entry of it's truth table is true), and that it is the unique odd function in its family.

A seal is also a periodic set partition. The members of its family shall be called its blocks.

seal 1001 0000 0110 0000   (Ж 4471)

This seal has adicity 4, and is shown with (the lowest possible) arity 4, i.e. with a truth table of length 16.
The 16×16 matrix on the left is its family matrix.
It also describes a partition of the set into four blocks, which are shown in different colors.
On the right the set partition is shown as a vertex coloring of the tesseract graph.   (The 4×4 matrix shows essentially the same.)

The weight of a Boolean function is be the quotient of the sum and the length of its truth table.
The weight of a seal is , where is its depth.
The unique seal with depth 0 is the tautology. The seals with depth 1 are the negated variadic XORs with one or more arguments.

Occurences

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The fixed points of Walsh permutations are seals.
The symmetry of any Boolean function is the symmetry of a seal.

Equivalence classes

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Equivalence classes (EC) of seals are factions. Those of blocks are clans.
Thus an EC can the represented by the smallest Zhegalkin index of the faction or the clan.   (That of the faction is easier to calculate.)

Antipodes

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The seals with arity a form a symmetric Hasse diagram, whose top node is the tautology.
From top to bottom the layers are depth = 0...a. From bottom to top they are rank = 0...a. The number of seals in layer n is  .
For the given arity each seal has a Walsh spectrum. It's non-zero entries are   (the weight of the seal's finite truth table).
Their pattern describes another seal in the opposite layer of the Hasse diagram, which shall be called its antipode.
When a seal in EC x has an antipode in EC y, then all seals in x have antipodes in y. Each EC has an antipode for a given arity. (Some EC are their own antipodes.)

The following seal is the antipode of the example shown above. Their depth (and rank) is 2, and both are on the middle layer of the Hasse diagram.

Cohorts

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While a seal has different antipodes for each arity, they all start with the same binary pattern, and then continue with zeros.
In other words, all antipodes of a seal can be described with the same finite set of integers (which corresponds to an entry of sequence Rose).
This set of integers shall be called cohort. It is a property of Boolean functions in general – not just of seals.


Integer representations

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Integer sequences

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Quantities

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number pyramids

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The 🌊 wave pyramids Liana and Wisteria are refinements of triangle Oak.   In Liana the dimension valency is added. In Wisteria the dimension cohort weight.
Their second sides are the triangle Ash (for Liana) and its reflection (for Wisteria).

The 💧 drop pyramids Ivy and Lonicera are refinements of triangle Maple.   In Ivy the dimension valency is added. In Lonicera the dimension cohort weight.
Their second sides are the triangles Aspen (for Ivy) and Alder (for Lonicera).

seals a×d×v: pyramids Liana and Ivy

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seals a×d×c: pyramids Wisteria and Lonicera

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