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 with the property that . More formally, . 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. However this latter term may have other meanings in other areas of mathematics.
Properties In a p-algebra L, for all The map is antitone. In particular, and . The map is a closure. . . . .
The set is called the skeleton of L. S(L) is a -subsemilattice of L and together with forms a Boolean algebra (the complement in this algebra is ). In general, S(L) is not a sublattice of L. In a distributive p-algebra, S(L) is the set of complemented elements of L.
Every element x with the property (or equivalently, ) is called dense. Every element of the form is dense. D(L), the set of all the dense elements in L is a filter of L. A distributive p-algebra is Boolean if and only if .
Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.
Examples Every finite distributive lattice is pseudocomplemented. Every Stone algebra is pseudocomplemented. In fact, a Stone algebra can be defined as a pseudocomplemented distributive lattice L in which any of the following equivalent statements hold for all S(L) is a sublattice of L;
. Every Heyting algebra is pseudocomplemented. If X is a topological space, the (open set) topology on X is a pseudocomplemented (and distributive) lattice with the meet and join being the usual union and intersection of open sets. The pseudocomplement of an open set A is the interior of the set complement of A. Furthermore, the dense elements of this lattice are exactly the dense open subsets in the topological sense.
Relative pseudocomplement A relative pseudocomplement of a with respect to b is a maximal element c such that . This binary operation is denoted . A lattice with a relative pseudocomplement for each pair of elements is called an 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 could be defined using relative pseudocomplement as .
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