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 ${\displaystyle x^{*}\in L}${\displaystyle x^{*}\in L} with the property that ${\displaystyle x\wedge x^{*}=0}${\displaystyle x\wedge x^{*}=0}. More formally, ${\displaystyle x^{*}=\max\{y\in L\mid x\wedge y=0\}}${\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.^{[1] [2]} However this latter term may have other meanings in other areas of mathematics.

In a p-algebra L, for all ${\displaystyle x,y\in L:}${\displaystyle x,y\in L:}^{[1] [2]}

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

The set ${\displaystyle S(L){\stackrel {\mathrm {d} ef}{=}}\{x^{*}\mid x\in L\}}${\displaystyle S(L){\stackrel {\mathrm {d} ef}{=}}\{x^{*}\mid x\in L\}} is called the skeleton of L. S(L) is a ${\displaystyle \wedge }${\displaystyle \wedge }-subsemilattice of L and together with ${\displaystyle x\cup y=(x\vee y)^{**}=(x^{*}\wedge y^{*})^{*}}${\displaystyle x\cup y=(x\vee y)^{**}=(x^{*}\wedge y^{*})^{*}} forms a Boolean algebra (the complement in this algebra is ${\displaystyle ^{*}}${\displaystyle ^{*}}).^{[1] [2]} 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 ${\displaystyle x^{*}=0}${\displaystyle x^{*}=0} (or equivalently, ${\displaystyle x^{**}=1}${\displaystyle x^{**}=1}) is called dense. Every element of the form ${\displaystyle x\vee x^{*}}${\displaystyle x\vee x^{*}} is dense. D(L), the set of all the dense elements in L is a filter of L.^{[1] [2]} A distributive p-algebra is Boolean if and only if ${\displaystyle D(L)=\{1\}}${\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 ${\displaystyle x,y\in L:}${\displaystyle x,y\in L:}^[1]
  • S(L) is a sublattice of L;
  • ${\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}}${\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}};
  • ${\displaystyle (x\vee y)^{**}=x^{**}\vee y^{**}}${\displaystyle (x\vee y)^{**}=x^{**}\vee y^{**}};
  • ${\displaystyle x^{*}\vee x^{**}=1}${\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 ${\displaystyle a\wedge c\leq b}${\displaystyle a\wedge c\leq b}. This binary operation is denoted ${\displaystyle a\to b}${\displaystyle 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 ${\displaystyle a^{*}}${\displaystyle a^{*}} could be defined using relative pseudocomplement as ${\displaystyle a\to 0}${\displaystyle a\to 0}.^[4]

• Topological vector lattice

• Lattice theory