Editing Field of sets (section)

== Fields of sets with additional structure ==

=== Sigma algebras and measure spaces ===

If an algebra over a set is closed under countable [[Union (set theory)|unions]] (hence also under [[countable]] [[Intersection (set theory)|intersections]]), it is called a [[sigma algebra]] and the corresponding field of sets is called a '''measurable space'''. The complexes of a measurable space are called '''measurable sets'''. The [[Lynn Harold Loomis|Loomis]]-[[Roman Sikorski|Sikorski]] theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called '''abstract sigma algebras''') and measurable spaces.

A '''measure space''' is a triple <math>( X, \mathcal{F}, \mu  )</math> where <math>( X, \mathcal{F} )</math> is a measurable space and <math>\mu</math> is a [[Measure theory|measure]] defined on it. If <math>\mu</math> is in fact a [[Probability theory|probability measure]] we speak of a '''probability space''' and call its underlying measurable space a '''sample space'''. The points of a sample space are called '''sample points''' and represent potential outcomes while the measurable sets (complexes) are called '''events''' and represent properties of outcomes for which we wish to assign probabilities. (Many use the term '''sample space''' simply for the underlying set of a probability space, particularly in the case where every subset is an event.) Measure spaces and probability spaces play a foundational role in [[measure theory]] and [[probability theory]] respectively.

In applications to [[Physics]] we often deal with measure spaces and probability spaces derived from rich mathematical structures such as [[inner product space]]s or [[topological group]]s which already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes.

=== Topological fields of sets ===

A '''topological field of sets''' is a triple <math>( X, \mathcal{T}, \mathcal{F} )</math> where <math>( X, \mathcal{T} )</math> is a [[topological space]] and <math>( X, \mathcal{F} )</math> is a field of sets which is closed under the [[closure operator]] of <math>\mathcal{T}</math> or equivalently under the [[interior operator]] i.e. the closure and interior of every complex is also a complex. In other words, <math>\mathcal{F}</math> forms a subalgebra of the power set [[interior algebra]] on <math>( X, \mathcal{T} ).</math>

Topological fields of sets play a fundamental role in the representation theory of interior algebras and [[Heyting algebra]]s. These two classes of algebraic structures provide the [[Algebraic semantics (mathematical logic)|algebraic semantics]] for the [[modal logic]] ''S4'' (a formal mathematical abstraction of [[Epistemic|epistemic logic]]) and [[intuitionistic logic]] respectively. Topological fields of sets representing these algebraic structures provide a related topological [[Semantics of logic|semantics]] for these logics.

Every interior algebra can be represented as a topological field of sets with the underlying Boolean algebra of the interior algebra corresponding to the complexes of the topological field of sets and the interior and closure operators of the interior algebra corresponding to those of the topology. Every [[Heyting algebra]] can be represented by a topological field of sets with the underlying lattice of the Heyting algebra corresponding to the lattice of complexes of the topological field of sets that are open in the topology. Moreover the topological field of sets representing a Heyting algebra may be chosen so that the open complexes generate all the complexes as a Boolean algebra. These related representations provide a well defined mathematical apparatus for studying the relationship between truth modalities (possibly true vs necessarily true, studied in modal logic) and notions of provability and refutability (studied in intuitionistic logic) and is thus deeply connected to the theory of [[modal companion]]s of [[intermediate logic]]s.

Given a topological space the [[Topology glossary|clopen]] sets trivially form a topological field of sets as each clopen set is its own interior and closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets, however in general the topology of a topological field of sets can differ from the topology generated by taking arbitrary unions of complexes and in general the complexes of a topological field of sets need not be open or closed in the topology.

==== Algebraic fields of sets and Stone fields ====

A topological field of sets is called '''algebraic''' if and only if there is a base for its topology consisting of complexes.

If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology.

Topological fields of sets that are separative, compact and algebraic are called '''Stone fields''' and provide a generalization of the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the [[Interior algebra#Open and closed elements|open elements]] of the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - the '''Stone representation'''. (The topology of the Stone representation is also known as the '''McKinsey–Tarski Stone topology''' after the mathematicians who first generalized Stone's result for Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology).

=== 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.

=== Complex algebras and fields of sets on relational structures===

The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbitrary (normal) [[Boolean algebra with operators|Boolean algebras with operators]]. For this we consider structures <math>( X, (R_i)_I, \mathcal{F} ) </math> where <math>( X,(R_i)_I ) </math> is a [[relational structure]] i.e. a set with an indexed family of [[Relation (mathematics)|relation]]s defined on it, and <math>( X, \mathcal{F} ) </math> is a field of sets. The '''complex algebra''' (or '''algebra of complexes''') determined by a field of sets <math>\mathbf{X} = ( X, \left(R_i\right)_I, \mathcal{F} ) </math> on a relational structure, is the Boolean algebra with operators
<math display="block">\mathcal{C}(\mathbf{X}) = ( \mathcal{F}, \cap, \cup, \prime, \empty, X, (f_i)_I )</math>
where for all <math>i \in I,</math> if <math>R_i</math> is a relation of arity <math>n + 1,</math> then <math>f_i</math> is an operator of arity <math>n</math> and for all <math>S_1, \ldots, S_n \in \mathcal{F}</math>
<math display="block">f_i(S_1, \ldots, S_n) = \left\{ x \in X : \text{ there exist } x_1 \in S_1, \ldots, x_n \in S_n \text{ such that } R_i(x_1, \ldots, x_n, x) \right\}</math>

This construction can be generalized to fields of sets on arbitrary [[algebraic structure]]s having both [[Operation (mathematics)|operators]] and relations as operators can be viewed as a special case of relations. If <math>\mathcal{F}</math> is the whole power set of <math>X</math> then <math>\mathcal{C}(\mathbf{X})</math> is called a '''full complex algebra''' or '''power algebra'''.

Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is [[Isomorphism|isomorphic]] to the complex algebra corresponding to the field.

(Historically the term '''complex''' was first used in the case where the algebraic structure was a [[Group (mathematics)|group]] and has its origins in 19th century [[group theory]] where a subset of a group was called a '''complex'''.)