linguistic:paraphrasing, stylistic:style_transfer
Complements and pseudo-complements

Let L {\displaystyle L} be a bounded lattice with greatest element 1 and least element 0. Two elements x {\displaystyle x} and y {\displaystyle y} of L {\displaystyle L} are complements of each other if and only if:

In general, some elements of a bounded lattice might not have a complement, and others might have more than one complement. For example, the set { 0, 1 / 2, 1 } {\displaystyle \{0,1/2,1\}} with its usual ordering is a bounded lattice, and 1 2 {\displaystyle {\tfrac {1}{2}}} does not have a complement. In the bounded lattice N, the element a {\displaystyle a} has two complements, viz. b {\displaystyle b} and c {\displaystyle c} (see Pic. 11). A bounded lattice for which every element has a complement is called a complemented lattice.

A complemented lattice that is also distributive is a Boolean algebra. For a distributive lattice, the complement of x, {\displaystyle x,} when it exists, is unique.

In the case that the complement is unique, we write ¬ x = y {\textstyle \lnot x=y} and equivalently, ¬ y = x. {\textstyle \lnot y=x.} The corresponding unary operation over L, {\displaystyle L,} called complementation, introduces an analogue of logical negation into lattice theory.

Heyting algebras are an example of distributive lattices where some members might be lacking complements. Every element z {\displaystyle z} of a Heyting algebra has, on the other hand, a pseudo-complement, also denoted ¬ x. {\textstyle \lnot x.} The pseudo-complement is the greatest element y {\displaystyle y} such that x ∧ y = 0. {\displaystyle x\wedge y=0.} If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.