that contains the empty set as an element, and is closed under the operations of taking complements in
Fields of sets play an essential role in the representation theory of Boolean algebras.
forms a subalgebra of the power set Boolean algebra of
is called a σ-algebra if the following additional condition (4) is satisfied: For an arbitrary set
One notable consequence: the number of complexes, if finite, is always of the form
This power set representation can be constructed more generally for any complete atomic Boolean algebra.
This leads us to construct a representation of a Boolean algebra by taking its set of ultrafilters and forming complexes by associating with each element of the Boolean algebra the set of ultrafilters containing that element.
It is the basis of Stone's representation theorem for Boolean algebras and an example of a completion procedure in order theory based on ideals or filters, similar to Dedekind cuts.
(The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the top elements under these homomorphisms.)
These definitions arise from considering the topology generated by the complexes of a field of sets.
is the topology formed by taking arbitrary unions of complexes.
The area of mathematics known as Stone duality is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a duality exists between Boolean algebras and Boolean spaces.
If an algebra over a set is closed under countable unions (hence also under countable intersections), it is called a sigma algebra and the corresponding field of sets is called a measurable space.
The Loomis-Sikorski theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called abstract sigma algebras) and measurable spaces.
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.
In applications to Physics we often deal with measure spaces and probability spaces derived from rich mathematical structures such as inner product spaces or topological groups which already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes.
is a field of sets which is closed under the closure operator of
forms a subalgebra of the power set interior algebra on
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 companions of intermediate logics.
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 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.
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.
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 modal frames which are fields of sets with an additional accessibility relation providing representations of modal algebras.
A preorder field is called algebraic (or tight) if and only if it has a set of complexes
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.
This construction can be generalized to fields of sets on arbitrary algebraic structures having both operators and relations as operators can be viewed as a special case of relations.
Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra corresponding to the field.