Seal (discrete mathematics)



Seal is a neologism for a mathematical object, that is essentially a subgroup of addition. The addition of nimbers is the bitwise XOR of non-negative integers. For a finite set $$\{0,...,2^n-1\}$$ it forms the Boolean group 2n.

A seal shall be defined as a Boolean function whose family matrix is also the matrix of an. 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.

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 $$\frac{1}{2^d}$$, where $$d$$ is its depth. The unique seal with depth 0 is the tautology. The seals with depth 1 are the negated variadic s with one or more arguments.

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

The following triangle (left) shows the number of seals by arity (rows) and depth (columns). The triangle on the right shows the corresponding numbers of clans.

E.g. there are (4, 2) = 35 4-ary seals of depth 2, and they fall into (4, 2) = 6 different clans.

One may be interested in all Boolean functions in the seal families. Their number for arity n is (n).