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Field of sets
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==== Algebraic and canonical preorder fields ==== A preorder field is called '''algebraic''' (or '''tight''') if and only if it has a set of complexes <math>\mathcal{A}</math> which determines the preorder in the following manner: <math>x \leq y</math> if and only if for every complex <math>S \in \mathcal{A}</math>, <math>x \in S</math> implies <math>y \in S</math>. The preorder fields obtained from ''S4'' theories are always algebraic, the complexes determining the preorder being the sets of possible worlds in which the sentences of the theory closed under necessity hold. A separative compact algebraic preorder field is said to be '''canonical'''. Given an interior algebra, by replacing the topology of its Stone representation with the corresponding [[Specialization (pre)order|canonical preorder]] (specialization preorder) we obtain a representation of the interior algebra as a canonical preorder field. By replacing the preorder by its corresponding [[Alexandrov topology#Duality with preordered sets|Alexandrov topology]] we obtain an alternative representation of the interior algebra as a topological field of sets. (The topology of this "'''Alexandrov representation'''" is just the [[Alexandrov topology#Categorical description of the duality|Alexandrov bi-coreflection]] of the topology of the Stone representation.) While representation of modal algebras by general modal frames is possible for any normal modal algebra, it is only in the case of interior algebras (which correspond to the modal logic ''S4'') that the general modal frame corresponds to topological field of sets in this manner.
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