Editing Interior algebra (section)

==Stone duality and representation for interior algebras==
[[Stone duality]] provides a category theoretic duality between Boolean algebras and a class of topological spaces known as [[Boolean space]]s. Building on nascent ideas of relational semantics (later formalized by [[Saul Kripke|Kripke]]) and a result of R. S. Pierce, [[Bjarni Jónsson|Jónsson]], [[Alfred Tarski|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 [[Field of sets#Complex algebras and fields of sets on relational structures|power set construction]]. In the case of interior algebras the interior (or closure) operator corresponds to a pre-order on the Boolean space. Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as '''pseudo-epimorphisms''' or '''p-morphisms''' for short. This generalization of Stone duality to interior algebras based on the Jónsson–Tarski representation was investigated by Leo Esakia and is also known as the ''Esakia duality for S4-algebras (interior algebras)'' and is closely related to the [[Esakia duality]] for Heyting algebras.

Whereas the Jónsson–Tarski generalization of Stone duality applies to Boolean algebras with operators in general, the connection between interior algebras and topology allows for another method of generalizing Stone duality that is unique to interior algebras. An intermediate step in the development of Stone duality is [[Stone's representation theorem for Boolean algebras|Stone's representation theorem]], which represents a Boolean algebra as a [[field of sets]]. The Stone topology of the corresponding Boolean space is then generated using the field of sets as a [[topological basis]]. Building on the [[topological semantics]] introduced by Tang Tsao-Chen for Lewis's modal logic, [[J.C.C. McKinsey|McKinsey]] and Tarski showed that by generating a topology equivalent to using only the complexes that correspond to open elements as a basis, a representation of an interior algebra is obtained as a [[Field of sets#topological field of sets|topological field of sets]]—a field of sets on a topological space that is closed with respect to taking interiors or closures. 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).

The pre-order obtained in the Jónsson–Tarski approach corresponds to the accessibility relation in the Kripke semantics for an S4 theory, while the intermediate field of sets corresponds to a representation of the Lindenbaum–Tarski algebra for the theory using the sets of possible worlds in the Kripke semantics in which sentences of the theory hold. Moving from the field of sets to a Boolean space somewhat obfuscates this connection. By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and Kripke frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics. The pre-order can be obtained as the specialization pre-order of the McKinsey–Tarski topology. The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates. Via a functor that instead replaces the pre-order with its corresponding Alexandrov topology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico-reflection of the McKinsey–Tarski topology. The approach of formulating a topological duality for interior algebras using both the Stone topology of the Jónsson–Tarski approach and the Alexandrov topology of the pre-order to form a bi-topological space has been investigated by G. Bezhanishvili, R.Mines, and P.J. Morandi. The McKinsey–Tarski topology of an interior algebra is the intersection of the former two topologies.