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 $x^{*}\in L$ 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.

Properties

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.

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 $x,y\in L:$
  • 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.

Relative pseudocomplement

A relative pseudocomplement of a with respect to b is a maximal element c such that $a\wedge c\le b$. This binary operation is denoted $a\to b$. 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 $a\to 0$.