(Boolean) function, BF usually meant as BF with infinite arity and periodic truth table similar to Boolean expression
truth table, TT usually meant as a truth table of finite length, determined by an arity
valency ≤ adicity ≤ arity
Valency is the number of arguments actually used. It is the number of circles in the Euler diagram.
Adicity follows from the biggest atom. 2adicity is the required TT length, or the period length of the infinite truth table.
The term arity is used in two slightly different ways:
arity n as an argument (e.g. of a class method) to get a finite truth table of length E.g. can be shown as 3-ary 0000 0011 or as 4-ary 0000 0011 0000 0011.
arity n as a shorthand for (as in: the 16 2-ary Boolean functions)
(For a while there may also be a third way, namely the erroneous use instead of valency or adicity.)
atom Atoms are also called sets or arguments of a BF. what is usually shown by a circle and labeled A, B, C...
atomvals the vector of atoms of a BF
root A root BF has no gaps before or between the atoms. Its valency and adicity are equal. often called non-degenerate
root and atomvals
A Boolean function is determined by its root and its atomvals.
It is a root, iff there are no gaps before or between the atoms. (The atomvals are consecutive integers starting from 0, or they are empty.)
Every BF has a root, but there is a small ambiguity:
as a BF: its equivalent with gapless atomvals
as a TT: binary vector of length 2valency without repeating patterns
spread not dense
segment geometric element of an Euler diagram, e.g. its cells and the walls between them The number of segments in a Venn diagram is 3valency.
dimension of a segment
A segment has a dimension, namely the number of zeros in its ternary label.
(From the perspective of Euler diagrams the term is counter-intuitive, because it corresponds to the hypercube, rather than to the cross polytope. For the hypersplits it is used in the same way.)
The relationships between segments that differ in only one digit are important:
Another segment with a 0 in the differing place is a superior. (Points are superior to lines, lines to areas, etc.)
Another segment with + or − in the differing place is an inferior.
Another segment with the opposite sign in the differing place is a neighbor.
blighted arity can be reduced bloated or blotted (blight, blightless)
bloated some arguments are equal or complementary to each other (bloat, bloatless)
blotted some arguments are equal or complementary to niverse or empty set (blot, blotless)
transformation signed permutation that turns elements of the same clan into each other
clan negation and permutation equivalence class partitioned into families and factions
family negation equivalence class
faction permutation equivalence class
cluster A cluster contains four or eight factions, that are complements and twins. (category)
(Zhegalkin) twin Zhegalkin index interpreted as TT of the same length (E.g. all bits true and only left bit true are always twins, because the Zhegalkin index of the tautology is 1.)