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.^[1][2] However this latter term may have other meanings in other areas of mathematics.

Properties

In a p-algebra L, for all ${\displaystyle x,y\in L:}$^[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; indeed, so do pseudocomplemented semilattices.^[3]

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 ${\displaystyle x,y\in L:}$^[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 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.^[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, 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]

See also

• Topological vector lattice

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. Balbes, Raymond; Horn, Alfred (September 1970). "Stone Lattices". Duke Math. J. 37 (3): 537–545. doi:10.1215/S0012-7094-70-03768-3.
4. Birkhoff, Garrett (1973). Lattice Theory (3rd ed.). AMS. p. 44.