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 <math>x^*\in L</math> with the property that <math>x\wedge x^*=0</math>. More formally, <math>x^* = \max\{y\in L\mid x\wedge y = 0 \}</math>. 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'''.<ref name="Blyth">{{cite book|author=T.S. Blyth|title=Lattices and Ordered Algebraic Structures|year=2006|publisher=Springer Science & Business Media|isbn=978-1-84628-127-3|at=Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119}}</ref><ref name="Bergman">{{cite book|author=Clifford Bergman|title=Universal Algebra: Fundamentals and Selected Topics|year=2011|publisher=CRC Press|isbn=978-1-4398-5129-6|pages=63–70}}</ref> However this latter term may have other meanings in other areas of mathematics.
==Properties== In a ''p''-algebra ''L'', for all <math>x, y \in L:</math><ref name="Blyth"/><ref name="Bergman"/> * The map <math>x \mapsto x^*</math> is antitone. In particular, <math>0^* = 1</math> and <math>1^* = 0</math>. * The map <math>x \mapsto x^{**}</math> is a closure. * <math>x^* = x^{***}</math>. * <math>(x\vee y)^* = x^* \wedge y^*</math>. * <math>(x\wedge y)^{**} = x^{**} \wedge y^{**}</math>. * <math>x\wedge(x\wedge y)^* = x\wedge y^*</math>.
The set <math>S(L) \stackrel{\mathrm def}{=} \{ x^* \mid x\in L \}</math> is called the '''skeleton''' of ''L''. ''S''(''L'') is a <math>\wedge</math>-subsemilattice of ''L'' and together with <math>x\cup y = (x\vee y)^{**} = (x^*\wedge y^*)^*</math> forms a Boolean algebra (the complement in this algebra is <math>^*</math>).<ref name="Blyth"/><ref name="Bergman"/> In general, ''S''(''L'') is not a sublattice of ''L''.<ref name="Bergman"/> In a distributive ''p''-algebra, ''S''(''L'') is the set of complemented elements of ''L''.<ref name="Blyth"/>
Every element ''x'' with the property <math>x^* = 0</math> (or equivalently, <math>x^{**} = 1</math>) is called '''dense'''. Every element of the form <math>x\vee x^*</math> is dense. ''D''(''L''), the set of all the dense elements in ''L'' is a filter of ''L''.<ref name="Blyth"/><ref name="Bergman"/> A distributive ''p''-algebra is Boolean if and only if <math>D(L) = \{1\}</math>.<ref name="Blyth"/>
Pseudocomplemented lattices form a variety; indeed, so do pseudocomplemented semilattices.<ref name="Balbes & Horn">{{cite journal|last1=Balbes|first1=Raymond|last2=Horn|first2=Alfred|author2link = Alfred Horn|date=September 1970|title=Stone Lattices|journal=Duke Math. J.|volume=37|issue=3|pages=537–545|doi=10.1215/S0012-7094-70-03768-3}}</ref>
== Examples == * Every finite distributive lattice is pseudocomplemented.<ref name="Blyth"/> * 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 <math>x, y \in L:</math><ref name="Blyth"/> ** ''S''(''L'') is a sublattice of ''L''; ** <math>(x\wedge y)^* = x^*\vee y^*</math>; ** <math>(x\vee y)^{**} = x^{**}\vee y^{**}</math>; ** <math>x^* \vee x^{**} = 1</math>. * Every Heyting algebra is pseudocomplemented.<ref name="Blyth"/> * 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.<ref name="Bergman"/>
== Relative pseudocomplement == A '''relative pseudocomplement''' of ''a'' with respect to ''b'' is a maximal element ''c'' such that <math>a\wedge c\le b</math>. This binary operation is denoted <math>a\to b</math>. A lattice with a relative pseudocomplement for each pair of elements is called an '''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 <math>a^*</math> could be defined using relative pseudocomplement as <math>a\to 0</math>.<ref>{{cite book |last1= Birkhoff |first1= Garrett |author-link1= Garrett Birkhoff | title = Lattice Theory | edition= 3rd|year= 1973 |publisher= AMS |page=44}}</ref>
== See also ==
* {{annotated link|Topological vector lattice}}
== References ==
{{reflist}}
Category:Lattice theory