In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a latticeL with bottom element 0, an element x ∈ L is said to have a pseudocomplement if there exists a greatest elementx* ∈ L, disjoint from x, with the property that x ∧ x* = 0. More formally, x* = max. The lattice L itself is called a pseudocomplemented lattice if every element ofL is pseudocomplemented. Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition, 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 x, y ∈ L:
The map x ↦ x* is antitone. In particular, 0* = 1 and 1* = 0.
The set S ≝ is called the skeleton of L. S is a ∧-subsemilattice of L and together with x ∪ y = ** = * forms a Boolean algebra. In general, S is not a sublattice of L. In a distributive p-algebra, S is the set of complementedelements of L. Every element x with the property x* = 0 is called dense. Every element of the form x ∨ x* is dense. D, the set of all the dense elements in L is a filter of L. A distributive p-algebra is Boolean if and only ifD =. Pseudocomplemented lattices form a variety.
Every Stone algebra is pseudocomplemented. In fact, a Stone algebra can be defined as a pseudocomplemented distributive latticeL in which any of the following equivalent statements hold for all x, y ∈ L:
A relative pseudocomplement of a with respect to b is a maximal elementc 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 case, an implicative lattice may not have a minimal element, if such element exists, then pseudocomplement a* could be defined using relative pseudocomplement as a → 0.
