Editing Interior algebra (section)

=== 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'''.