Properties of Boolean functions/soft

Soft properties depend on the Template:Boolf-prop 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 $A\wedge B$:

• arity 3: Template:Spaces $(A\wedge B)\oplus C$ Template:Spaces 0001 0001 XOR 0000 1111 = 0001 1110
• arity 4: Template:Spaces $(A\wedge B)\oplus D$ Template:Spaces 0001 0001 0001 0001 XOR 0000 0000 1111 1111 = 0001 0001 1110 1110

Template:Anchorultra 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.

Template:Boolf-prop and Template:Boolf-props can be merged with their complements, which creates Template:Boolf-prop-families and super-clans.
Further merging them with their Template:Boolf-props creates ultra-families and ultra-clans.
This family is a complete ultra-family: 1100 1010 Template:Spaces (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 Template:Boolf-prop.
Template:Boolf-prop BF belong to tribes denoted by their Template:Boolf-prop. All Template:Boolf-prop BF belong to a tribe on their own.

Template:Anchorprincipalities and dominions

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