In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element x ∈ L is said to have a pseudocomplement if there exists a greatest element x* ∈ L with the property that x ∧ x* = 0. More formally, x* = max. The lattice L itself is called a pseudocomplemented lattice if every element of L is pseudocomplemented. Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a p-algebra.[1] [2] However this latter term may have other meanings in other areas of mathematics.
In a p-algebra L, for all
x,y\inL:
The set S(L) ≝ is called the skeleton of L. S(L) is a ∧-subsemilattice of L and together with x ∪ y = (x∨y)** = (x* ∧ y*)* forms a Boolean algebra (the complement in this algebra is *).[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. Every element of the form x ∨ x* is 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]
Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.[3]
x,y\inL:
A relative pseudocomplement of a with respect to b is a maximal element c such that a∧c≤b. This binary operation is denoted a→b. A lattice with the pseudocomplement for each two elements is called implicative lattice, or Brouwerian lattice. In general, an implicative lattice may not have a minimal element. If such a minimal element exists, then each pseudocomplement a* could be defined using relative pseudocomplement as a → 0.[4]