Editing Interior algebra (section)

=== 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''''')).