In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement.
[1][2] In general, S(L) is not a sublattice of L.[2] In a distributive p-algebra, S(L) is the set of complemented elements of L.[1] Every element x with the property x* = 0 (or equivalently, x** = 1) is called dense.
D(L), the set of all the dense elements in L is a filter of L.[1][2] A distributive p-algebra is Boolean if and only if D(L) = {1}.
[3] A relative pseudocomplement of a with respect to b is a maximal element c such that a∧c≤b.
In general, an implicative lattice may not have a minimal element.