Pseudocomplement

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 ${\displaystyle x^{*}\in L}$ with the property that ${\displaystyle x\wedge x^{*}=0}$. More formally, ${\displaystyle x^{*}=\max\{y\in L\mid x\wedge y=0\}}$. 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.¹² However this latter term may have other meanings in other areas of mathematics.

In a p-algebra L, for all ${\displaystyle x,y\in L:}$¹²

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

The set ${\displaystyle S(L){\stackrel {\mathrm {d} ef}{=}}\{x^{*}\mid x\in L\}}$ is called the skeleton of L. S(L) is a ${\displaystyle \wedge }$-subsemilattice of L and together with ${\displaystyle x\cup y=(x\vee y)^{**}=(x^{*}\wedge y^{*})^{*}}$ forms a Boolean algebra (the complement in this algebra is ${\displaystyle ^{*}}$).¹² 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 ${\displaystyle x^{*}=0}$ (or equivalently, ${\displaystyle x^{**}=1}$) is called dense. Every element of the form ${\displaystyle x\vee x^{*}}$ is dense. D(L), the set of all the dense elements in L is a filter of L.¹² A distributive p-algebra is Boolean if and only if ${\displaystyle D(L)=\{1\}}$.¹

Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.³

• Every finite distributive lattice is pseudocomplemented.¹
• 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 ${\displaystyle x,y\in L:}$¹
  • S(L) is a sublattice of L;
  • ${\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}}$;
  • ${\displaystyle (x\vee y)^{**}=x^{**}\vee y^{**}}$;
  • ${\displaystyle x^{*}\vee x^{**}=1}$.
• Every Heyting algebra is pseudocomplemented.¹
• 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.²

A relative pseudocomplement of a with respect to b is a maximal element c such that ${\displaystyle a\wedge c\leq b}$. This binary operation is denoted ${\displaystyle a\to b}$. 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 ${\displaystyle a^{*}}$ could be defined using relative pseudocomplement as ${\displaystyle a\to 0}$.⁴