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Pseudocomplement

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Source updated April 3, 2026

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 $x^{*}\in L$ with the property that $x\wedge x^{*}=0$ . More formally, $x^{*}=\max\{y\in L\mid x\wedge y=0\}$ . 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.

01Properties

In a p-algebra L, for all $x,y\in L:$

The map $x\mapsto x^{*}$ is antitone. In particular, $0^{*}=1$ and $1^{*}=0$ .
The map $x\mapsto x^{**}$ is a closure.
$x^{*}=x^{***}$ .
$(x\vee y)^{*}=x^{*}\wedge y^{*}$ .
$(x\wedge y)^{**}=x^{**}\wedge y^{**}$ .
$x\wedge (x\wedge y)^{*}=x\wedge y^{*}$ .

The set $S(L){\stackrel {\mathrm {d} ef}{=}}\{x^{*}\mid x\in L\}$ is called the skeleton of L. S(L) is a $\wedge$ -subsemilattice of L and together with $x\cup y=(x\vee y)^{**}=(x^{*}\wedge y^{*})^{*}$ 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 $x^{*}=0$ (or equivalently, $x^{**}=1$ ) is called dense. Every element of the form $x\vee x^{*}$ 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 $D(L)=\{1\}$ .

Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.

02Examples

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;
- $(x\wedge y)^{*}=x^{*}\vee y^{*}$ ;
- $(x\vee y)^{**}=x^{**}\vee y^{**}$ ;
- $x^{*}\vee x^{**}=1$ .
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.

03Relative pseudocomplement

A relative pseudocomplement of a with respect to b is a maximal element c such that $a\wedge c\leq b$ . This binary operation is denoted $a\to b$ . 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 $a^{*}$ could be defined using relative pseudocomplement as $a\to 0$ .

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Sources and credits

This article is adapted from the Wikipedia article “Pseudocomplement”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Pseudocomplement

01Properties

02Examples

03Relative pseudocomplement

Sources and credits

Related topics

Topological vector lattice