Soft properties depend on the arity of the BF.

twin

The Zhegalkin twin of a BF is its Zhegalkin index interpreted as a truth table of length 2^arity.

half-complement

The half-complement of a BF is its XOR with x_arity−1.   See e.g. the half-complements of ${\displaystyle A\land B}$:

• arity 3:       ${\displaystyle (A\land B)\oplus C}$       0001 0001 XOR 0000 1111 = 0001 1110
• arity 4:       ${\displaystyle (A\land B)\oplus D}$       0001 0001 0001 0001 XOR 0000 0000 1111 1111 = 0001 0001 1110 1110

ultra equivalence classes

The two families on the left form a super-family. The family on the right is a super-family on its own. Together these two super-families form an ultra-family.

Families and clans can be merged with their complements, which creates super-families and super-clans.
Further merging them with their half-complements creates ultra-families and ultra-clans.
This family is a complete ultra-family: 1100 1010     (So its matrix of truth tables is mirror symmetric.)
(Factions do not have a unique half-complement, so there are no ultra-factions.)

tribe

For a given arity the tribe is the next biggest equivalence class after the ultra-clan.
Blunt BF belong to tribes denoted by their consul weight. All sharp BF belong to a tribe on their own.

tribes for arities 1, 2, 3

1-ary truth tables Walsh spectra consuls (A253315) tribe 0 1 2 3 [0 0] [1 0] [0 1] [1 1] [ 0 0] [ 1 1] [ 1 -1] [ 2 0] [0 0] [0 0] [0 1] [0 1] 0 0 1 1 0 s s 1 2-ary truth tables Walsh spectra consuls (A253315) tribes sharp: white 0: dark blue 1: light blue 2: green Calculated with this Python script 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [0 0 0 0] [1 0 0 0] [0 1 0 0] [1 1 0 0] [0 0 1 0] [1 0 1 0] [0 1 1 0] [1 1 1 0] [0 0 0 1] [1 0 0 1] [0 1 0 1] [1 1 0 1] [0 0 1 1] [1 0 1 1] [0 1 1 1] [1 1 1 1] [ 0 0 0 0] [ 1 1 1 1] [ 1 -1 1 -1] [ 2 0 2 0] [ 1 1 -1 -1] [ 2 2 0 0] [ 2 0 0 -2] [ 3 1 1 -1] [ 1 -1 -1 1] [ 2 0 0 2] [ 2 -2 0 0] [ 3 -1 1 1] [ 2 0 -2 0] [ 3 1 -1 1] [ 3 -1 -1 -1] [ 4 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 1] [0 1 0 1] [0 0 1 1] [0 0 1 1] [0 1 1 0] [0 1 1 0] [0 1 1 0] [0 1 1 0] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 0 1] [0 0 0 0] [0 0 0 0] 0 0 1 1 2 2 3 3 3 3 2 2 1 1 0 0 0 s s 1 s 1 2 s s 2 1 s 1 s s 0 3-ary

principalities and dominions

A principality is a set of n-ary BF whose (n+1)-ary noble equivalents form a faction. Dominions are closely related.

partner and friend

The partner/friend of a BF differs from its truth table in the least/most significant bit.
E.g. 0110 has the partner 1110 and the friend 0111.

squad, platoon and company

squad: union of a super-faction and its partner   (always two super-factions, i.e. four factions)
platoon: union of a squad and its twin   (one or two squads, i.e. four or eight factions)
company: union of a platoon and its friend   (one or two platoons)

images for arity 3
See overview on Commons.