Editing Interior algebra (section)

== Open and closed elements ==
Elements of an interior algebra satisfying the condition ''x''<sup>I</sup> = ''x'' are called '''[[open set|open]]'''. The [[complement (order theory)|complements]] of open elements are called '''[[closed set|closed]]''' and are characterized by the condition ''x''<sup>C</sup> = ''x''. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called '''[[regular open set|regular open]]''' and closures of open elements are called '''regular closed'''. Elements that are both open and closed are called '''[[clopen set|clopen]]'''. 0 and 1 are clopen.

An interior algebra is called '''Boolean''' if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of '''trivial''' interior algebras, which are the single element interior algebras characterized by the identity 0 = 1.