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In mathematics a field of sets is a pair  where X is a set and 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 forms a subalgebra of the power set Boolean algebra of X. (Many authors refer to itself as a field of sets.) Elements of X are called points and those of are called complexes. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the...
In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...
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. In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...
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. An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0. ...
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ...
An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0. ...
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. In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
In mathematical order theory, an ideal is a special subset of a partially ordered set. ...
In mathematics, a filter is a special subset of a partially ordered set. ...
In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards...
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. In abstract algebra, a homomorphism is a structure-preserving map. ...
Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...
Separative and compact fields of sets: towards Stone duality - A field of sets is called separative 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 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  the complexes form a base for a topology, we denote the corresponding topological space by  . Then In mathematics, a filter is a special subset of a partially ordered set. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...
- is always a zero-dimensional space.
- is a Hausdorff space if and only if
 is separative. - is a compact space with compact open sets if and only if is compact.
- is a Boolean space with clopen sets if and only if is both separative and compact.
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. In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ...
In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Boolean spaces, i. ...
In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ...
In mathematics, duality has numerous meanings. ...
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. In mathematics the term countable set is used to describe the size of a set, e. ...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...
A measure space is a triple  where is a measurable space and μ is a measure defined on it. If μ 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. In mathematics, a measure is a function that assigns a number, e. ...
It has been suggested that this article or section be merged with Probability axioms. ...
In mathematics, a measure is a function that assigns a number, e. ...
It has been suggested that this article or section be merged with Probability axioms. ...
Topological fields of sets A topological field of sets is a triple  where  is a topological space and is a field of sets which is closed under the closure operator of  or equivalently under the interior operator i.e. the closure and interior of every complex is also a complex. In other words forms a subalgebra of the power set interior algebra on . Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: x ≤ C(x) for all x, i. ...
In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called...
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. This is a glossary of some terms used in the branch of mathematics known as 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 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. In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called...
Preorder fields A preorder field is a triple  where  is a preordered set and is a field of sets. In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered 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 In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open. ...
 there exists a  with  and there exists a with  for all  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. In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ...
Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ...
1. ...
In mathematical logic, the Lindenbaum-Tarski algebra A of a logical theory T consists of the equivalence classes of sentences p of the theory, under the equivalence relation ~ defined by p ~ q when p and q are logically equivalent in T. That is, in T q can be deduced from...
Algebraic and canonical preorder fields A preorder field is called algebraic if and only if it has a set of complexes  which determines the preorder in the following manner:  if and only if for every complex  ,  implies . 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.) In the branch of mathematics known as topology the specialization (or canonical) preorder defines a preorder on the set of the points of a topological space. ...
In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open. ...
In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open. ...
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  where  is a relational structure i.e. a set with an indexed family of relations defined on it, and is a field of sets. The complex algebra (or algebra of complexes) determined by a field of sets on a relational structure, is the Boolean algebra with operators In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
where for all , if is a relation of arity n + 1, then is an operator of arity n and for all - there exist such that
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 is the whole power set of then is called a full complex algebra or power algebra. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
(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.) This picture illustrates how the hours in a clock form a group. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
See also Algebra of sets George Boole Boolean algebra Boolean function Boolean logic Boolean homomorphism Boolean Implicant Boolean prime ideal theorem Boolean-valued model Boolean satisfiability problem Booles syllogistic canonical form (Boolean algebra) compactness theorem Complete Boolean algebra connective -- see logical operator de Morgans laws Augustus De Morgan duality (order...
In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...
In mathematics, a measure is a function that assigns a number, e. ...
It has been suggested that this article or section be merged with Probability axioms. ...
In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called...
In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open. ...
In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i. ...
In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ...
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, 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
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