Field of sets

"Set algebra" redirects here. For the basic properties and laws of sets, see Algebra of sets.

In mathematics a field of sets is a pair $\langle X, \mathcal{F} \rangle$ where $X$ is a set and $\mathcal{F}$ is an algebra over $X$ i.e., a non-empty subset of the power set of $X$ closed under the intersection and union of pairs of sets and under complements of individual sets. In other words $\mathcal{F}$ forms a subalgebra of the power set Boolean algebra of $X$ . (Many authors refer to $\mathcal{F}$ itself as a field of sets. The word "field" in "field of sets" is not used with the meaning of field from field theory.) Elements of $X$ are called points and those of $\mathcal{F}$ are called complexes and are said to be the admissible sets of $X$ .

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.

Fields of sets in the representation theory of Boolean algebras

Stone representation

Every finite Boolean algebra can be represented as a whole 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 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.

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.

Separative and compact fields of sets: towards Stone duality

A field of sets is called separative (or differentiated) if and only if for every pair of distinct points there is a complex containing one and not the other.

A field of sets is called compact if and only if for every proper filter over $X\$ the intersection of all the complexes contained in the filter is non-empty.

These definitions arise from considering the topology generated by the complexes of a field of sets. Given a field of sets $\mathbf{X}= \langle X, \mathcal{F} \rangle$ the complexes form a base for a topology, we denote the corresponding topological space by $T(\mathbf{X})$ . Then

$T(\mathbf{X})$ is always a zero-dimensional space.
$T(\mathbf{X})$ is a Hausdorff space if and only if $\mathbf{X}$ is separative.
$T(\mathbf{X})$ is a compact space with compact open sets $\mathcal{F}$ if and only if $\mathbf{X}$ is compact.
$T(\mathbf{X})$ is a Boolean space with clopen sets $\mathcal{F}$ if and only if $\mathbf{X}$ is both separative and compact (in which case it is described as being descriptive)

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.

Fields of sets with additional structure

Sigma algebras and measure spaces

If an algebra over a set is closed under countable intersections and countable unions, 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.

A measure space is a triple $\langle X, \mathcal{F}, \mu \rangle$ where $\langle X, \mathcal{F} \rangle$ is a measurable space and $\mu$ is a measure defined on it. If $\mu$ is in fact a 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 samples 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.

The Loomis-Sikorski theorem provides a Stone-type duality between abstract sigma algebras and measurable spaces.

Topological fields of sets

A topological field of sets is a triple $\langle X, \mathcal{T}, \mathcal{F} \rangle$ where $\langle X, \mathcal{T} \rangle$ is a topological space and $\langle X, \mathcal{F} \rangle$ is a field of sets which is closed under the closure operator of $\mathcal{T}$ or equivalently under the interior operator i.e. the closure and interior of every complex is also a complex. In other words $\mathcal{F}$ forms a subalgebra of the power set interior algebra on $\langle X, \mathcal{T} \rangle$ .

Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space.

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.

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 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.

Preorder fields

A preorder field is a triple $\langle X, \leq , \mathcal{F} \rangle$ where $\langle X, \leq \rangle$ is a preordered set and $\langle X, \mathcal{F}\rangle$ 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

\mbox{Int}(S) = \{x \in X :

there exists a

y \in S

with

y \leq x \}

and

\mbox{Cl}(S) = \{ x \in X :

there exists a

y \in S

with

x \leq y \}

for all

S \in \mathcal{F}

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 (a formal mathematical abstraction of epistemic logic), 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.

Algebraic and canonical preorder fields

A preorder field is called algebraic if and only if it has a set of complexes $\mathcal{A}$ which determines the preorder in the following manner: $x \leq y$ if and only if for every complex $S \in \mathcal{A}$ , $x \in S$ implies $y \in S$ . 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 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.)

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 algebras with operators. For this we consider structures $\langle X, ( R_i )_I, \mathcal{F} \rangle$ where $\langle X, ( R_i )_I \rangle$ is a relational structure i.e. a set with an indexed family of relations defined on it, and $\langle X, \mathcal{F} \rangle$ is a field of sets. The complex algebra (or algebra of complexes) determined by a field of sets $\mathbf{X} = \langle X, ( R_i )_I, \mathcal{F} \rangle$ on a relational structure, is the Boolean algebra with operators

\mathcal{C}(\mathbf{X}) = \langle \mathcal{F}, \cap, \cup, \prime, \empty, X, ( f_i )_I \rangle

where for all $i \in I$ , if $R_i\$ is a relation of arity $n+1$ , then $f_i\$ is an operator of arity $n$ and for all $S_1,...,S_n \in \mathcal{F}$

f_i(S_1,...,S_n) = \{ x \in X :

there exist

x_1 \in S_1 ,...,x_n \in S_n

such that

R_i(x_1,...,x_n,x) \}\

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. If $\mathcal{F}$ is the whole power set of $X\$ then $\mathcal{C}(\mathbf{X})$ 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 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.)

References

Goldblatt, R., Algebraic Polymodal Logic: A Survey, Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450, July 2000
Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, 44, p. 173-242, 1989
Johnstone, Peter T. (1982). Stone spaces (3rd edition ed.). Cambridge: Cambridge University Press. ISBN 0-521-33779-8.
Naturman, C.A., Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Mathematics, 1991
Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed., Handbook of Modal Logic, Volume 3 of Studies in Logic and Practical Reasoning, Elsevier, 2006

External links

Hazewinkel, Michiel, ed. (2001), "Algebra of sets", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4