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; ( x ∧ y ) ∗ = x ∗ ∨ y ∗ {\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}}; ( x ∨ y ) ∗ ∗ = x ∗ ∗ ∨ y ∗ ∗ {\displaystyle (x\vee y)^{**}=x^{**}\vee y^{**}}; x ∗ ∨ x ∗ ∗ = 1 {\displaystyle 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 ∧ c ≤ b {\displaystyle a\wedge c\leq b}. This binary operation is denoted a → b {\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 a ∗ {\displaystyle a^{*}} could be defined using relative pseudocomplement as a → 0 {\displaystyle a\to 0}.

See also