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, disjoint from x, 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.[1][2] However this latter term may have other meanings in other areas of mathematics.

Properties

In a p-algebra L, for all x, y ∈ L:[1][2]

  • 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 *).[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 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.[1][2] A distributive p-algebra is Boolean if and only if D(L) = {1}.[1]

Pseudocomplemented lattices form a variety.[2]

Examples
  • 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 x, y ∈ L:[1]
    • S(L) is a sublattice of L;
    • (x∧y)* = x* ∨ y*;
    • (x∨y)** = x** ∨ y**;
    • x* ∨ x** = 1.
  • Every Heyting algebra is pseudocomplemented.[1]
  • If X is a set, 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]

Relative pseudocomplement

A relative pseudocomplement of a with respect to b is a maximal element c such that a∧c≤b. This binary operation is denoted a→b. A lattice with the pseudocomplement for each two elements is called implicative lattice, or Brouwerian lattice. In general case, an implicative lattice may not have a minimal element, if such element exists, then pseudocomplement a* could be defined using relative pseudocomplement as a → 0.[3]

References
  1. T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3.
  2. Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6.
  3. Birkhoff, Garrett (1973). Lattice Theory (3rd ed.). AMS. p. 44.
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