Field of sets - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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 Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) 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 In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In other words $\mathcal{F}$ forms a subalgebra of the power set Boolean algebra of $X$ . In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. (Many authors refer to $\mathcal{F}$ itself as a field of sets. ) Elements of $X$ are called points and those of $\mathcal{F}$ are called complexes.

Fields of sets play an essential role in the representation theory of Boolean algebras. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Every Boolean algebra can be represented as a field of sets.

1 Fields of sets in the representation theory of Boolean algebras
- 1.1 Stone representation
- 1.2 Separative and compact fields of sets: towards Stone duality
2 Fields of sets with additional structure
3 See also
4 References

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). In the mathematical field of Order theory, given two elements a and b of a Partially ordered set, one says that b covers a This power set representation can be constructed more generally for any complete atomic Boolean algebra. This article is about a type of Mathematical structure. For the notion from Computer science, see Complete Boolean algebra (computer science. In the mathematical field of Order theory, given two elements a and b of a Partially ordered set, one says that b covers a

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. In the mathematical field of Set theory, an ultrafilter on a set X is a collection of Subsets of X that is a filter, that 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, Stone's representation theorem for Boolean algebras states that every Boolean algebra is Isomorphic to a Field of sets. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In mathematical Order theory, an ideal is a special subset of a Partially ordered set (poset 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 into two non-empty

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. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector (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. A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional

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 $X\$ the intersection of all the complexes contained in the filter is non-empty. In Mathematics, a filter is a special Subset of a Partially ordered set.

These definitions arise from considering the topology generated by the complexes of a field of sets. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. 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 $T(\mathbf{X})$ . In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets Then

$T(\mathbf{X})$ is always a zero-dimensional space. In Mathematics, the Lebesgue covering dimension or topological dimension of a Topological space is defined to be the minimum value of n, such
$T(\mathbf{X})$ is a Hausdorff space if and only if $\mathbf{X}$ is separative. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space
$T(\mathbf{X})$ 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 Mathematics, there is an ample supply of categorical dualities between certain categories of Topological spaces and categories of Partially ordered

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. In Mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is Isomorphic to a Field of sets. 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, there is an ample supply of categorical dualities between certain categories of Topological spaces and categories of Partially ordered In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution.

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. 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 In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty 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 $μ$ is a measure defined on it. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with If $μ$ is in fact a probability measure we speak of a probability space and call its underlying measurable space a sample space. Probability theory is the branch of Mathematics concerned with analysis of random phenomena 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 the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Probability theory is the branch of Mathematics concerned with analysis of random phenomena

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. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " 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 $\langle X, \mathcal{T} \rangle$ . In Abstract algebra, an interior algebra is a certain type of Algebraic structure that encodes the idea of the topological Interior of a set

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. This is a glossary of some terms used in the branch of Mathematics known as Topology. 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). In Abstract algebra, an interior algebra is a certain type of Algebraic structure that encodes the idea of the topological Interior of a set 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. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions

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 Topology, an Alexandrov space (or Alexandrov-discrete space) is a Topological space in which the intersection of any family of Open sets 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. A modal logic is any system of formal logic that attempts to deal with modalities. Kripke semantics (also known as relational semantics or frame semantics, and often confused with Possible world semantics) is a formal Semantics Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge In Mathematical logic, the Lindenbaum-Tarski algebra A of a logical theory T consists of the Equivalence classes of sentences

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. In the branch of Mathematics known as Topology, the specialization (or canonical) preorder is a natural Preorder on the set of the By replacing the preorder by its corresponding Alexandrov topology we obtain an alternative representation of the interior algebra as a topological field of sets. In Topology, an Alexandrov space (or Alexandrov-discrete space) is a Topological space in which the intersection of any family of Open sets (The topology of this "Alexandrov representation" is just the Alexandrov bi-coreflection of the topology of the Stone representation. In Topology, an Alexandrov space (or Alexandrov-discrete space) is a Topological space in which the intersection of any family of Open sets )

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. In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations e. a set with an indexed family of relations defined on it, and $\langle X, \mathcal{F} \rangle$ is a field of sets. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations 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. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, an operator is a function which operates on (or modifies another function 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

(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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. )

References

Goldblatt, R. This is a list of topics around Boolean algebra and propositional logic. The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Probability theory is the branch of Mathematics concerned with analysis of random phenomena In Abstract algebra, an interior algebra is a certain type of Algebraic structure that encodes the idea of the topological Interior of a set In Topology, an Alexandrov space (or Alexandrov-discrete space) is a Topological space in which the intersection of any family of Open sets In Mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is Isomorphic to a Field of sets. In Mathematics, there is an ample supply of categorical dualities between certain categories of Topological spaces and categories of Partially ordered In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that , 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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