# 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]

## 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

- 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. - Clifford Bergman (2011).
*Universal Algebra: Fundamentals and Selected Topics*. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. - Birkhoff, Garrett (1973).
*Lattice Theory*(3rd ed.). AMS. p. 44.