Editing Interior algebra (section)

== Relationships to other areas of mathematics ==

=== Topology ===
Given a [[topological space]] '''''X''''' = ⟨''X'', ''T''⟩ one can form the [[power set]] Boolean algebra of ''X'':

:{{math|1=⟨''P''(''X''), ∩, ∪, ′, ø, ''X''⟩}}

and extend it to an interior algebra

:{{math|1='''''A'''''('''''X''''') = ⟨''P''(''X''), ∩, ∪, ′, ø, ''X'', <sup>I</sup>⟩}},

where <sup>I</sup> is the usual topological interior operator. For all ''S'' ⊆ ''X'' it is defined by

:{{math|1=''S''<sup>I</sup> = ∪ {{mset|''O'' | ''O'' ⊆ ''S'' and ''O'' is open in '''''X'''''}}}}

For all ''S'' ⊆ ''X'' the corresponding closure operator is given by

:{{math|1=''S''<sup>C</sup> = ∩ {{mset|''C'' | ''S'' ⊆ ''C'' and ''C'' is closed in '''''X'''''}}}}

''S''<sup>I</sup> is the largest open subset of ''S'' and ''S''<sup>C</sup> is the smallest closed superset of ''S'' in '''''X'''''. The open, closed, regular open, regular closed and clopen elements of the interior algebra '''''A'''''('''''X''''') are just the open, closed, regular open, regular closed and clopen subsets of '''''X''''' respectively in the usual topological sense.

Every [[Completeness (order theory)|complete]] [[Atomic (order theory)|atomic]] interior algebra is [[isomorphism|isomorphic]] to an interior algebra of the form '''''A'''''('''''X''''') for some [[topological space]] '''''X'''''.  Moreover, every interior algebra can be [[embedding|embedded]] in such an interior algebra giving a representation of an interior algebra as a '''[[Field_of_sets#Topological_fields_of_sets|topological field of sets]]'''. The properties of the structure '''''A'''''('''''X''''') are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called '''topo-Boolean algebras''' or '''topological Boolean algebras'''.

Given a [[continuous map]] between two topological spaces

:{{math|1=''f''&nbsp;:&nbsp;'''''X'''''&nbsp;→&nbsp;'''''Y'''''}}

we can define a [[completeness (order theory)|complete]] topomorphism

:{{math|1='''''A'''''(''f'')&nbsp;:&nbsp;'''''A'''''('''''Y''''')&nbsp;→&nbsp;'''''A'''''('''''X''''')}}

by

:'''''A'''''(''f'')(''S'') = ''f''<sup>−1</sup>[''S'']

for all subsets ''S'' of '''''Y'''''. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If '''Top''' is the [[category of topological spaces]] and continuous maps and '''Cit''' is the [[category theory|category]] of complete atomic interior algebras and complete topomorphisms then '''Top''' and '''Cit''' are [[Dual (category theory)|dually isomorphic]] and {{math|1='''''A'''''&nbsp;:&nbsp;'''Top'''&nbsp;→&nbsp;'''Cit'''}} is a [[functor|contravariant functor]] that is a dual isomorphism of categories. '''''A'''''(''f'') is a homomorphism if and only if ''f'' is a continuous [[open map]].

Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties:

*'''''X''''' is [[empty set|empty]] if and only if '''''A'''''('''''X''''') is trivial
*'''''X''''' is [[indiscrete space|indiscrete]] if and only if '''''A'''''('''''X''''') is [[simple algebra|simple]]
*'''''X''''' is [[discrete space|discrete]] if and only if '''''A'''''('''''X''''') is Boolean
*'''''X''''' is [[almost discrete space|almost discrete]] if and only if '''''A'''''('''''X''''') is [[semisimple algebraic group|semisimple]]
*'''''X''''' is [[Alexandrov topology|finitely generated]] (Alexandrov) if and only if '''''A'''''('''''X''''') is '''operator complete''' i.e. its interior and closure operators distribute over arbitrary meets and joins respectively
*'''''X''''' is [[connected space|connected]] if and only if '''''A'''''('''''X''''') is [[directly indecomposable]]
*'''''X''''' is [[ultraconnected space|ultraconnected]] if and only if '''''A'''''('''''X''''') is [[finitely subdirectly irreducible]]
*'''''X''''' is [[compact space|compact]] ultra-connected if and only if '''''A'''''('''''X''''') is [[subdirectly irreducible]]

==== Generalized topology ====

The modern formulation of topological spaces in terms of [[topological space|topologies]] of open subsets, motivates an alternative formulation of interior algebras: A '''generalized topological space''' is an [[algebraic structure]] of the form

:⟨''B'', ·, +, ′, 0, 1, ''T''⟩

where ⟨''B'', ·, +, ′, 0, 1⟩ is a Boolean algebra as usual, and ''T'' is a unary relation on ''B'' (subset of ''B'') such that:

#{{math|1=0,1 ∈ ''T''}}
#''T'' is closed under arbitrary joins (i.e. if a join of an arbitrary subset of ''T'' exists then it will be in ''T'')
#''T'' is closed under finite meets
#For every element ''b'' of ''B'', the join {{math|1=Σ{{mset|''a'' ∈''T'' | ''a'' ≤ ''b''}}}} exists

''T'' is said to be a '''generalized topology''' in the Boolean algebra.

Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space

:⟨''B'', ·, +, ′, 0, 1, ''T''⟩

we can define an interior operator on ''B'' by {{math|1=''b''<sup>I</sup> = Σ{{mset|''a'' ∈''T'' | ''a'' ≤ ''b''}}}} thereby producing an interior algebra whose open elements are precisely ''T''. Thus generalized topological spaces are equivalent to interior algebras.

Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added relations, so that standard results from [[universal algebra]] apply.

==== Neighbourhood functions and neighbourhood lattices ====
The topological concept of [[Neighbourhood (mathematics)|neighbourhood]]s can be generalized to interior algebras: An element ''y'' of an interior algebra is said to be a '''neighbourhood''' of an element ''x'' if {{math|1=''x'' ≤ ''y''<sup>I</sup>}}. The set of neighbourhoods of ''x'' is denoted by ''N''(''x'') and forms a [[Filter (mathematics)|filter]]. This leads to another formulation of interior algebras:

A '''neighbourhood function''' on a Boolean algebra is a mapping ''N'' from its underlying set ''B'' to its set of filters, such that:

#For all {{math|1=''x'' ∈ ''B'', max{{mset|''y'' ∈ ''B'' | ''x'' ∈ ''N''(''y'')}}}} exists
#For all {{math|1=''x'',''y'' ∈ ''B'', ''x'' ∈ ''N''(''y'')}} if and only if there is a {{math|1=''z'' ∈ ''B''}} such that {{math|1=''y'' ≤ ''z'' ≤ ''x''}} and {{math|1=''z'' ∈ ''N''(''z'')}}.

The mapping ''N'' of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function ''N'' on a Boolean algebra with underlying set ''B'', we can define an interior operator by {{math|1=''x''<sup>I</sup> = max{{mset|y ∈ ''B'' | ''x'' ∈ ''N''(''y'')}}}} thereby obtaining an interior algebra. {{tmath|1=N(x)}} will then be precisely the filter of neighbourhoods of ''x'' in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.

In terms of neighbourhood functions, the open elements are precisely those elements ''x'' such that {{math|1=''x'' ∈ ''N''(''x'')}}. In terms of open elements {{math|1=''x'' ∈ ''N''(''y'')}} if and only if there is an open element ''z'' such that {{math|1=''y'' ≤ ''z'' ≤ ''x''}}.

Neighbourhood functions may be defined more generally on [[semilattice|(meet)-semilattice]]s producing the structures known as [[neighbourhood lattice|neighbourhood (semi)lattice]]s. Interior algebras may thus be viewed as precisely the '''Boolean neighbourhood lattices''' i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.

=== Modal logic<!--'S4 algebra' and 'Lewis algebra' redirect here--> ===
Given a [[theory (logic)|theory]] (set of formal sentences) ''M'' in the modal logic '''S4''', we can form its [[Lindenbaum–Tarski algebra]]:

:'''''L'''''(''M'') = ⟨''M'' / ~, ∧, ∨, ¬, ''F'', ''T'', □⟩

where ~ is the equivalence relation on sentences in ''M'' given by ''p'' ~ ''q'' if and only if ''p'' and ''q'' are [[Logical equivalence|logically equivalent]] in ''M'', and ''M'' / ~ is the set of equivalence classes under this relation. Then '''''L'''''(''M'') is an interior algebra. The interior operator in this case corresponds to the [[modal logic|modal operator]] □ ('''necessarily'''), while the closure operator corresponds to ◊ ('''possibly'''). This construction is a special case of a more general result for [[modal algebra]]s and modal logic.

The open elements of '''''L'''''(''M'') correspond to sentences that are only true if they are '''necessarily''' true, while the closed elements correspond to those that are only false if they are '''necessarily''' false.

Because of their relation to '''S4''', interior algebras are sometimes called '''S4 algebras'''<!--boldface per WP:R#PLA--> or '''Lewis algebras'''<!--boldface per WP:R#PLA-->, after the [[philosophical logic|logician]] [[Clarence Irving Lewis|C. I. Lewis]], who first proposed the modal logics '''S4''' and '''S5'''.

=== Preorders ===
Since interior algebras are (normal) [[Boolean algebra (structure)|Boolean algebra]]s with [[unary operation|operators]], they can be represented by [[field of sets|fields of sets]] on appropriate relational structures. In particular, since they are [[modal algebra]]s, they can be represented as [[field of sets|fields of sets]] on a set with a single [[binary relation]], called a [[Kripke frame]]. The Kripke frames corresponding to interior algebras are precisely the [[Preorder|preordered sets]]. [[Preorder|Preordered sets]] (also called ''S4-frames'') provide the [[Kripke semantics]] of the modal logic '''S4''', and the connection between interior algebras and preorders is deeply related to their connection with modal logic.

Given a [[Preorder|preordered set]] '''''X''''' = ⟨''X'', «⟩ we can construct an interior algebra

: {{math|1='''''B'''''('''''X''''') = ⟨''P''(''X''), ∩, ∪, ′, ø, ''X'', <sup>I</sup>⟩}}

from the [[power set]] [[Boolean algebra (structure)|Boolean algebra]] of ''X'' where the interior operator <sup>I</sup> is given by

:{{math|1=''S''<sup>I</sup> = {{mset|''x'' ∈ ''X'' | for all ''y'' ∈ ''X'', ''x'' « ''y'' implies ''y'' ∈ ''S''}}}} for all ''S'' ⊆ ''X''.

The corresponding closure operator is given by

:{{math|1=''S''<sup>C</sup> = {{mset|''x'' ∈ ''X'' | there exists a ''y'' ∈ ''S'' with ''y'' « ''x''}}}} for all ''S'' ⊆ ''X''.

''S''<sup>I</sup> is the set of all ''worlds'' inaccessible from ''worlds'' outside ''S'', and ''S''<sup>C</sup> is the set of all ''worlds'' accessible from some ''world'' in ''S''. Every interior algebra can be [[embedding|embedded]] in an interior algebra of the form '''''B'''''('''''X''''') for some [[Preorder|preordered set]] '''''X''''' giving the above-mentioned representation as a [[field of sets]] (a '''preorder field''').

This construction and representation theorem is a special case of the more general result for [[modal algebra]]s and Kripke frames. In this regard, interior algebras are particularly interesting because of their connection to [[topology]]. The construction provides the [[Preorder|preordered set]] '''''X''''' with a [[topological space|topology]], the [[Alexandrov topology]], producing a [[topological space]] '''''T'''''('''''X''''') whose open sets are:

:{{math|1={{mset|''O'' ⊆ ''X'' | for all ''x'' ∈ ''O'' and all ''y'' ∈ ''X'', ''x'' « ''y'' implies ''y'' ∈ ''O''}}}}.

The corresponding closed sets are:

:{{math|1={{mset|''C'' ⊆ ''X'' | for all ''x'' ∈ ''C'' and all ''y'' ∈ ''X'', ''y'' « ''x'' implies ''y'' ∈ ''C''}}}}.

In other words, the open sets are the ones whose ''worlds'' are inaccessible from outside (the '''up-sets'''), and the closed sets are the ones for which every outside ''world'' is inaccessible from inside (the '''down-sets'''). Moreover, '''''B'''''('''''X''''') = '''''A'''''('''''T'''''('''''X''''')).

=== Monadic Boolean algebras ===
Any [[monadic Boolean algebra]] can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic Boolean algebras are then precisely the [[Variety (universal algebra)|variety]] of interior algebras satisfying the identity ''x''<sup>IC</sup> = ''x''<sup>I</sup>. In other words, they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the [[Semisimple algebra|semisimple]] interior algebras. They are also the interior algebras corresponding to the modal logic '''S5''', and so have also been called '''S5 algebras'''.

In the relationship between preordered sets and interior algebras they correspond to the case where the preorder is an [[equivalence relation]], reflecting the fact that such preordered sets provide the Kripke semantics for '''S5'''. This also reflects the relationship between the [[monadic logic]] of quantification (for which monadic Boolean algebras provide an [[Lindenbaum–Tarski algebra|algebraic description]]) and '''S5''' where the modal operators □ ('''necessarily''') and ◊ ('''possibly''') can be interpreted in the Kripke semantics using monadic universal and existential quantification, respectively, without reference to an accessibility relation.

=== Heyting algebras ===
The open elements of an interior algebra form a [[Heyting algebra]] and the closed elements form a [[duality (order theory)|dual]] Heyting algebra. The regular open elements and regular closed elements correspond to the [[Pseudocomplement|pseudo-complemented]] elements and [[duality (order theory)|dual]] pseudo-complemented elements of these algebras respectively and thus form Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every [[Heyting algebra]] can be represented as the open elements of an interior algebra and the latter may be chosen to be an interior algebra generated by its open elements—such interior algebras correspond one-to-one with Heyting algebras (up to isomorphism) being the free Boolean extensions of the latter.

Heyting algebras [[Lindenbaum–Tarski algebra|play the same role]] for [[intuitionistic logic]] that interior algebras play for the modal logic '''S4''' and [[Boolean algebra (structure)|Boolean algebra]]s play for [[propositional logic]]. The relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and '''S4''', in which one can interpret theories of intuitionistic logic as '''S4''' theories [[deductive closure|closed]] under [[logical truth|necessity]]. The one-to-one correspondence between Heyting algebras and interior algebras generated by their open elements reflects the correspondence between extensions of intuitionistic logic and [[normal modal logic|normal]] extensions of the modal logic '''S4.Grz'''.

=== Derivative algebras ===
Given an interior algebra '''''A''''', the closure operator obeys the axioms of the [[Derivative algebra (abstract algebra)|derivative operator]], <sup>D</sup>. Hence we can form a [[Abstract algebra|derivative algebra]] '''''D'''''('''''A''''') with the same underlying Boolean algebra as '''''A''''' by using the closure operator as a derivative operator.

Thus interior algebras are [[Derivative algebra (abstract algebra)|derivative algebras]]. From this perspective, they are precisely the [[variety (universal algebra)|variety]] of derivative algebras satisfying the identity ''x''<sup>D</sup> ≥ ''x''. Derivative algebras provide the appropriate [[Lindenbaum–Tarski algebra|algebraic semantics]] for the modal logic '''wK4'''. Hence derivative algebras stand to topological [[derived set (mathematics)|derived set]]s and '''wK4''' as interior/closure algebras stand to topological interiors/closures and '''S4'''.
 
Given a derivative algebra '''''V''''' with derivative operator <sup>D</sup>, we can form an interior algebra {{math|1='''''I'''''('''''V''''')}} with the same underlying Boolean algebra as '''''V''''', with interior and closure operators defined by {{math|1=''x''<sup>I</sup> = ''x''·''x'' ′ <sup>D</sup> ′}} and {{math|1=''x''<sup>C</sup> = ''x'' + ''x''<sup>D</sup>}}, respectively. Thus every derivative algebra can be regarded as an interior algebra. Moreover, given an interior algebra '''''A''''', we have {{math|1='''''I'''''('''''D'''''('''''A''''')) = '''''A'''''}}. However, {{math|1='''''D'''''('''''I'''''('''''V''''')) = '''''V'''''}} does ''not'' necessarily hold for every derivative algebra '''''V'''''.