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. 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.

Properties

In a ''p''-algebra ''L'', for all $x, y \in L:$
* 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'') ≝ 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'') = .

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

Examples

* 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 $x, y \in L:$
** ''S''(''L'') is a sublattice of ''L'';
** (''x''∧''y'')* = ''x''* ∨ ''y''*;
** (''x''∨''y'')** = ''x''** ∨ ''y''**;
** ''x''* ∨ ''x''** = 1.
* Every Heyting algebra is pseudocomplemented.
* 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.

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.

See also

*

References

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Lattice theory