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 ∗ ∈ L {\displaystyle x^{*}\in L} with the property that x ∧ x ∗ = 0 {\displaystyle x\wedge x^{*}=0} . More formally, x ∗ = max { y ∈ L ∣ x ∧ 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. However this latter term may have other meanings in other areas of mathematics.
Properties In a p-algebra L, for all x , y ∈ L : {\displaystyle x,y\in L:}
The map x ↦ x ∗ {\displaystyle x\mapsto x^{*}} is antitone. In particular, 0 ∗ = 1 {\displaystyle 0^{*}=1} and 1 ∗ = 0 {\displaystyle 1^{*}=0} . The map x ↦ x ∗ ∗ {\displaystyle x\mapsto x^{**}} is a closure.
x ∗ = x ∗ ∗ ∗ {\displaystyle x^{*}=x^{***}} .
( x ∨ y ) ∗ = x ∗ ∧ y ∗ {\displaystyle (x\vee y)^{*}=x^{*}\wedge y^{*}} .
( x ∧ y ) ∗ ∗ = x ∗ ∗ ∧ y ∗ ∗ {\displaystyle (x\wedge y)^{**}=x^{**}\wedge y^{**}} .
x ∧ ( x ∧ y ) ∗ = x ∧ y ∗ {\displaystyle x\wedge (x\wedge y)^{*}=x\wedge y^{*}} . The set S ( L ) = d e f { x ∗ ∣ x ∈ 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 } -subsemilattice of L and together with x ∪ y = ( x ∨ y ) ∗ ∗ = ( x ∗ ∧ y ∗ ) ∗ {\displaystyle x\cup y=(x\vee y)^{**}=(x^{*}\wedge y^{*})^{*}} forms a Boolean algebra (the complement in this algebra is ∗ {\displaystyle ^{*}} ). 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 {\displaystyle x^{*}=0} (or equivalently, x ∗ ∗ = 1 {\displaystyle x^{**}=1} ) is called dense. Every element of the form x ∨ x ∗ {\displaystyle 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 } {\displaystyle 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 ∈ L : {\displaystyle x,y\in L:}
S(L) is a sublattice of L;
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