pseudocomplement

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

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

The set $S(L)\; \backslash stackrel\{\backslash mathrm\; def\}\{=\}\; \backslash \{\; x^*\; \backslash mid\; x\backslash in\; L\; \backslash \}$ is called the skeleton of L. S(L) is a $\backslash wedge$-subsemilattice of L and together with $x\backslash cup\; y\; =\; (x\backslash vee\; y)^\{**\}\; =\; (x^*\backslash wedge\; y^*)^*$ 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 $x\backslash vee\; x^*$ 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 $D(L)\; =\; \backslash \{1\backslash \}$.

Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.{{cite journal|last1=Balbes|first1=Raymond|last2=Horn|first2=Alfred|author2link = Alfred Horn|date=September 1970|title=Stone Lattices|journal=Duke Math. J.|volume=37|issue=3|pages=537–545|doi=10.1215/S0012-7094-70-03768-3}}

• 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 $x,\; y\; \backslash in\; L:$
• S(L) is a sublattice of L;
• $(x\backslash wedge\; y)^*\; =\; x^*\backslash vee\; y^*$;
• $(x\backslash vee\; y)^\{**\}\; =\; x^\{**\}\backslash vee\; y^\{**\}$;
• $x^*\; \backslash vee\; x^\{**\}\; =\; 1$.
• 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.

A relative pseudocomplement of a with respect to b is a maximal element c such that $a\backslash wedge\; c\backslash le\; b$. This binary operation is denoted $a\backslash to\; b$. 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 $a\backslash to\; 0$.{{cite book |last1= Birkhoff |first1= Garrett |author-link1= Garrett Birkhoff | title = Lattice Theory | edition= 3rd|year= 1973 |publisher= AMS |page=44}}

• {{annotated link|Topological vector lattice}}

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Category:Lattice theory