Field of sets

Algebraic concept in measure theory, also referred to as an algebra of sets.

Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.

Similarly, every finite Boolean algebra can be represented as a power set – the power set of its set of atoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power set representation can be constructed more generally for any complete atomic Boolean algebra.

In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Boolean algebra correspond to its ultrafilters and that an atom is below an element of a finite Boolean algebra if and only if that element is contained in the ultrafilter corresponding to the atom. 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. This construction does indeed produce a representation of the Boolean algebra as a field of sets and is known as the Stone representation. It is the basis of and an example of a completion procedure in order theory based on ideals or filters, similar to Dedekind cuts.

Alternatively one can consider the set of homomorphisms onto the two element Boolean algebra and form complexes by associating each element of the Boolean algebra with the set of such homomorphisms that map it to the top element. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the top elements under these homomorphisms.) With this approach one sees that Stone representation can also be regarded as a generalization of the representation of finite Boolean algebras by truth tables.

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space is known as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexes of the Stone representation. 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 complexes of a measurable space are called measurable sets. The Loomis-Sikorski theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called abstract sigma algebras) and measurable spaces.

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.

Topological fields of sets play a fundamental role in the representation theory of interior algebras and Heyting algebras. These two classes of algebraic structures provide the algebraic semantics for the modal logic S4 (a formal mathematical abstraction of epistemic logic) and intuitionistic logic respectively. Topological fields of sets representing these algebraic structures provide a related topological 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 companions of intermediate logics.

Given a topological space the 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.

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 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).

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 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 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 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 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.

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.

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