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* ∈ L with the property that x ∧ x* = 0. More formally, x* = max{ y ∈ L | x ∧ 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. In a p-algebra L, for all The map x ↦ x* is antitone. In particular, 0* = 1 and 1* = 0. The map x ↦ x** is a closure. x* = x***. (x∨y)* = x* ∧ y*. (x∧y)** = x** ∧ y**. The set S(L) ≝ { x** | x ∈ L } is called the skeleton of L. S(L) is a ∧-subsemilattice of L and together with x ∪ y = (x∨y)** = (x* ∧ 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 ∨ 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. 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 S(L) is a sublattice of L; (x∧y)* = x* ∨ y*; (x∨y)** = x** ∨ y**; x* ∨ x** = 1. Every Heyting algebra is pseudocomplemented.