Pseudocomplement Explained

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 xL is said to have a pseudocomplement if there exists a greatest element x* ∈ L with the property that xx* = 0. More formally, x* = max. 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\inL:

[1] [2]

The set S(L) ≝ is called the skeleton of L. S(L) is a ∧-subsemilattice of L and together with xy = (xy)** = (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 xx* 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]

Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.[3]

Examples

x,y\inL:

[1]

Relative pseudocomplement

A relative pseudocomplement of a with respect to b is a maximal element c such that acb. This binary operation is denoted ab. 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 → 0.[4]

Notes and References

  1. Book: T.S. Blyth. Lattices and Ordered Algebraic Structures. 2006. Springer Science & Business Media. 978-1-84628-127-3. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119.
  2. Book: Clifford Bergman. Universal Algebra: Fundamentals and Selected Topics. 2011. CRC Press. 978-1-4398-5129-6. 63–70.
  3. Balbes. Raymond. Horn. Alfred. Alfred Horn. September 1970. Stone Lattices. Duke Math. J.. 37. 3. 537–545. 10.1215/S0012-7094-70-03768-3.
  4. Book: Birkhoff . Garrett . Garrett Birkhoff . Lattice Theory . 3rd. 1973 . AMS . 44.