Studies of Euler diagrams/clans

Functions in the same NP equivalence class (clan) have Euler diagrams with the same shape.

The functions in it can be turned into each other with transformations. They negate and permute the arguments, i.e. they are signed permutations.

A diagram's symmetry determines the cardinality of the clan, i.e. the number of functions in it.

Let any function have $$k$$ transformations to itself. Then the diagram's symmetry group has $$k$$ elements, and the cardinality of the clan is $$\frac{2^n \cdot n!}{k}$$.

So the clans with the hightest possible cardinality of $$2^n \cdot n!$$ are those whose diagrams have no symmetry. For arity $$n = 3$$ that would be $$8 \cdot 6 = 48$$, but no such clan exists. (The hightest cardinality is 24.) For arity $$n = 4$$ that is $$16 \cdot 24 = 384$$. An example is the clan of. See table.

Apart from constants, multigrade XOR has the highest symmetry, and thus corresponds to the lowest cardinality of 2. An example is the that of. See table.

clan of
The diagrams are mirror symmetric, so the cardinality is 192. (All 382 diagrams are shown, i.e. both mirror images for each function.)

Compare the transformations between and.

tables
Each clan can be partitioned into smaller equivalence classes based only on permutation (factions) or negation (families). Within the conventions of this project, the factions are rows, while families partition the table in vertical stripes. (They shall be called columns.)

The following table shows the clan of  and its complement. (The complements make it an NPN-EC.)

Its contains 24 functions. The columns partition it into 3 families, each with 8 functions. The rows partition it into 6 factions. The four rows with mirror symmetric Venn diagrams contain 3, and the other two rows contain 6 functions. The families always have the same size, while that of the factions can vary.

The following tables show clans with only one family. They contain the 2-ary OR and the 3-ary OR :

The number of factions is at least 2. These contain the multigrade XOR and the complement of equality :