The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm[citation needed] following the work of Wim Blok.
Elements of an interior algebra satisfying the condition xI = x are called open.
The complements of open elements are called closed and are characterized by the condition xC = x.
A map f : A → B is a topomorphism if and only if f is a homomorphism between the Boolean algebras underlying A and B, that also preserves the open and closed elements of A.
Hence: (Such morphisms have also been called stable homomorphisms and closure algebra semi-homomorphisms.)
Sikorski thus defined a continuous homomorphism as a Boolean σ-homomorphism f between two σ-complete interior algebras such that f(x)C ≤ f(xC).
This definition had several difficulties: The construction acts contravariantly producing a dual of a continuous map rather than a generalization.
(Sikorski remarked on using non-σ-complete homomorphisms but included σ-completeness in his axioms for closure algebras.)
(C. Naturman returned to Sikorski's approach while dropping σ-completeness to produce topomorphisms as defined above.
For all S ⊆ X it is defined by For all S ⊆ X the corresponding closure operator is given by SI is the largest open subset of S and SC is the smallest closed superset of S in X.
Given a continuous map between two topological spaces we can define a complete topomorphism by for all subsets S of Y.
If Top is the category of topological spaces and continuous maps and Cit is the category of complete atomic interior algebras and complete topomorphisms then Top and Cit are dually isomorphic and A : Top → Cit is a contravariant functor that is a dual isomorphism of categories.
The Kripke frames corresponding to interior algebras are precisely the preordered sets.
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.
This construction and representation theorem is a special case of the more general result for modal algebras and Kripke frames.
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 closed under 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 extensions of the modal logic S4.Grz.
From this perspective, they are precisely the variety of derivative algebras satisfying the identity xD ≥ x.
However, D(I(V)) = V does not necessarily hold for every derivative algebra V. Stone duality provides a category theoretic duality between Boolean algebras and a class of topological spaces known as Boolean spaces.
Building on nascent ideas of relational semantics (later formalized by Kripke) and a result of R. S. Pierce, Jónsson, Tarski and G. Hansoul extended Stone duality to Boolean algebras with operators by equipping Boolean spaces with relations that correspond to the operators via a power set construction.
Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as pseudo-epimorphisms or p-morphisms for short.
By equipping topological fields of sets with appropriate morphisms known as field maps, C. Naturman showed that this approach can be formalized as a category theoretic Stone duality in which the usual Stone duality for Boolean algebras corresponds to the case of interior algebras having redundant interior operator (Boolean interior algebras).
R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and Kripke frames.
The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics.
The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates.