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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Stone's representation theorem for Boolean algebras states that every
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to a certain
field of sets In mathematics, a field of sets is a mathematical structure consisting of a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X called an algebra over X that contains the empty set as an element, and is closed unde ...
. The theorem is fundamental to the deeper understanding of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
of operators on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
.


Stone spaces

Each
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a (closed) basis consisting of all sets of the form \, where ''b'' is an element of ''B''. This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. For every Boolean algebra ''B'', ''S''(''B'') is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
; such spaces are called Stone spaces (also ''profinite spaces''). Conversely, given any topological space ''X'', the collection of subsets of ''X'' that are clopen (both closed and open) is a Boolean algebra.


Representation theorem

A simple version of Stone's representation theorem states that every Boolean algebra ''B'' is isomorphic to the algebra of clopen subsets of its Stone space ''S''(''B''). The isomorphism sends an element b \in B to the set of all ultrafilters that contain ''b''. This is a clopen set because of the choice of topology on ''S''(''B'') and because ''B'' is a Boolean algebra. Restating the theorem using the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
; the theorem states that there is a duality between the category of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s and the category of Stone spaces. This duality means that in addition to the correspondence between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra ''A'' to a Boolean algebra ''B'' corresponds in a natural way to a continuous function from ''S''(''B'') to ''S''(''A''). In other words, there is a contravariant functor that gives an equivalence between the categories. This was an early example of a nontrivial duality of categories. The theorem is a special case of
Stone duality In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they f ...
, a more general framework for dualities between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s and
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s. The proof requires either the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
or a weakened form of it. Specifically, the theorem is equivalent to the
Boolean prime ideal theorem In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by cons ...
, a weakened choice principle that states that every Boolean algebra has a prime ideal. An extension of the classical Stone duality to the category of Boolean spaces (= zero-dimensional locally compact Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor).


See also

* * * * * * * *


Citations


References

* * *{{cite book , last1=Burris , first1=Stanley N. , first2=H.P. , last2=Sankappanavar , title=A Course in Universal Algebra , publisher=Springer , date=1981 , isbn=3-540-90578-2 , url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html General topology Boolean algebra Theorems in lattice theory Categorical logic