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In [[mathematics]], '''Stone's representation theorem for Boolean algebras''' states that every [[Boolean algebra (structure)|Boolean algebra]] is [[isomorphic]] to a [[field of sets]]. The theorem is fundamental to the deeper understanding of [[Boolean logic|Boolean algebra]] that emerged in the first half of the 20th century. The theorem was first proved by [[Marshall H. Stone|Stone]] (1936), and thus named in his honor. Stone was led to it by his study of the [[spectral theory]] of [[linear operator|operators]] on a [[Hilbert space]].
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==Stone spaces==
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Each [[Boolean algebra (structure)|Boolean algebra]] ''B'' has an associated topological space, denoted here ''S''(''B''), called its '''Stone space'''. The points in ''S''(''B'') are the [[ultrafilter]]s 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 (topology)|basis]] consisting of all sets of the form
:<math>\{ x \in S(B) \mid b \in x\},</math>
where ''b'' is an element of ''B''.
 
For every Boolean algebra ''B'', ''S''(''B'') is a [[compact space|compact]] [[totally disconnected]] [[Hausdorff space|Hausdorff]] space; such spaces are called '''Stone spaces''' (also ''profinite spaces''). Conversely, given any topological space ''X'', the collection of subsets of ''X'' that are [[clopen set|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''&isin;''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]]; the theorem states that there is a [[duality of categories|duality]] between the [[category theory|category]] of [[Boolean algebra (structure)|Boolean algebra]]s  and the category of Stone spaces. This duality means that in addition to the isomorphisms 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 (category theory)|equivalence]] between the categories. This was an early example of a nontrivial duality of categories.
 
The theorem is a special case of [[Stone duality]], a more general framework for dualities between [[topological space]]s and [[partially ordered set]]s.
 
The proof requires either the [[axiom of choice]] or a weakened form of it. Specifically, the theorem is equivalent to the [[Boolean prime ideal theorem]], a weakened choice principle which states that every Boolean algebra has a prime ideal.
 
==See also==
* [[Field of sets]]
* [[List of Boolean algebra topics]]
* [[Stonean space]]
* [[Stone functor]]
* [[Profinite group]]
* [[Representation theorem]]
 
==References==
* [[Paul Halmos]], and Givant, Steven (1998) ''Logic as Algebra''. Dolciani Mathematical Expositions No. 21. [[The Mathematical Association of America]].
* [[Peter T. Johnstone|Johnstone, Peter T.]] (1982) ''Stone Spaces''. Cambridge University Press. ISBN 0-521-23893-5.
* [[Marshall H. Stone]] (1936) "[http://links.jstor.org/sici?sici=0002-9947%28193607%2940%3A1%3C37%3ATTORFB%3E2.0.CO%3B2-8 The Theory of Representations of Boolean Algebras,]" ''Transactions of the American Mathematical Society 40'': 37-111.
A monograph available free online:
* Burris, Stanley N., and H.P. Sankappanavar, H. P.(1981) ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]''  Springer-Verlag. ISBN 3-540-90578-2.
 
[[Category:General topology]]
[[Category:Boolean algebra]]
[[Category:Theorems in algebra]]
[[Category:Categorical logic]]

Latest revision as of 15:29, 25 September 2014

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