Seal (discrete mathematics)

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 ${\displaystyle \{0,...,2^{n}-1\}}$ 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 ${\displaystyle \{0\ldots 15\}}$ 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 ${\displaystyle {\frac {1}{2^{d}}}}$, where ${\displaystyle d}$ 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.

negated binary Walsh matrix
The seals with depth 1 are the positive rows of a negated binary Walsh matrix.   (The tautology with depth 0 is in the top row.)

The fixed points of Walsh permutations are seals.
The symmetry of Boolean functions is related to seals.

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 ${\displaystyle \mathrm {Oak} (a,n)}$.
For the given arity each seal has a Walsh spectrum. It's non-zero entries are ${\displaystyle 2^{rank}}$ (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 house x has an antipode in house y, then all seals in x have antipodes in y. Each house has an antipode for a given arity.

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.

seal 1000 1000 0001 0001   (Ж 1807)
The Walsh spectrum for arity 4 is (4, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0). Its non-zero entries form the pattern 1001 0000 0110 0000 of the antipode Ж 4471.

1001 0000 0000 1001 (Ж 1911) has cohort {0, 3, 12, 15}. So for arity 4 it is its own antipode. (See this file.)

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 as the same finite set of integers (which corresponds to an entry of sequence Rose).
This set of integers shall be called cohort. Every cohort belongs to a legion. Cohort and legion are properties of Boolean functions in general – not just of seals.

The cardinalities of cohort and legion are a powers of two.
The exponent for the cohort is depth. The exponent for the legion shall be called gravity.
Gravity is similar to valency. The pyramids TwistedLiana and Lonicera have it as one of their dimensions.