Editing Interior algebra (section)

=== 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.