Stone's representation theorem for Boolean algebras
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In
Stone spaces
Each
For every Boolean algebra B, S(B) is a
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 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
The theorem is a special case of Stone duality, a more general framework for dualities between topological spaces and partially ordered sets.
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 that states that every Boolean algebra has a prime ideal.
An extension of the classical Stone duality to the category of Boolean spaces (that is,
See also
- Stone's representation theorem for distributive lattices
- Representation theorem – Proof that every structure with certain properties is isomorphic to another structure
- Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
- List of Boolean algebra topics
- Stonean space– Topological space in which the closure of every open set is open
- Stone functor – Functor in category theory
- Profinite group – Topological group that is in a certain sense assembled from a system of finite groups
- Ultrafilter lemma– Maximal proper filter
Citations
References
- ISBN 0-88385-327-2.
- ISBN 0-521-23893-5.
- Burris, Stanley N.; Sankappanavar, H.P. (1981). A Course in Universal Algebra. Springer. ISBN 3-540-90578-2.