Editing Field of sets (section)

=== Preorder fields ===

A '''preorder field''' is a triple <math>( X, \leq , \mathcal{F} )</math> where <math>( X, \leq )</math> is a [[Preorder|preordered set]] and <math>( X, \mathcal{F} )</math> is a field of sets.

Like the topological fields of sets, preorder fields play an important role in the representation theory of interior algebras. Every interior algebra can be represented as a preorder field with its interior and closure operators corresponding to those of the [[Alexandrov topology]] induced by the preorder. In other words, for all <math>S \in \mathcal{F}</math>:
<math display="block">\mathrm{Int}(S) = \{ x \in X : \text{ there exists a } y \in S \text{ with } y \leq x \}</math> and
<math display="block">\mathrm{Cl}(S)  = \{ x \in X : \text{ there exists a } y \in S \text{ with } x \leq y \}</math> 

Similarly to topological fields of sets, preorder fields arise naturally in modal logic where the points represent the ''possible worlds'' in the [[Kripke semantics]] of a theory in the modal logic ''S4'', the preorder represents the accessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worlds in which individual sentences in the theory hold, providing a representation of the [[Lindenbaum–Tarski algebra]] of the theory. They are a special case of the [[General frame|general modal frames]] which are fields of sets with an additional accessibility relation providing representations of modal algebras.

==== 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.