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and the category of Stone spaces. This duality means that in addition to the correspondence between
Boolean algebras and their Stone spaces, each
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Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor).
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between the categories. This was an early example of a nontrivial duality of categories.
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An extension of the classical Stone duality to the category of
Boolean spaces (that is,
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that emerged in the first half of the 20th century. The theorem was first proved by
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or a weakened form of it. Specifically, the theorem is equivalent to the
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626:"The categories of Boolean lattices, Boolean rings and Boolean spaces"
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is isomorphic to the algebra of clopen subsets of its Stone space
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Every
Boolean algebra is isomorphic to a certain field of sets
369:. This is a clopen set because of the choice of topology on
101:. The theorem is fundamental to the deeper understanding of
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corresponds in a natural way to a continuous function from
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Stone's representation theorem for distributive lattices
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Pages displaying short descriptions of redirect targets
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of homomorphisms into the two-element
Boolean algebra.
345:
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679:. Dolciani Mathematical Expositions. Vol. 21.
591:"Some generalizations of the Stone Duality Theorem"
550:"The Theory of Representations of Boolean Algebras"
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Stone's representation theorem for
Boolean algebras
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242:
555:Transactions of the American Mathematical Society
443:, a more general framework for dualities between
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720:Burris, Stanley N.; Sankappanavar, H.P. (1981).
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384:Restating the theorem using the language of
365:to the set of all ultrafilters that contain
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307:). Conversely, given any topological space
109:. Stone was led to it by his study of the
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243:{\displaystyle \{x\in S(B)\mid b\in x\},}
69:Learn how and when to remove this message
32:This article includes a list of general
681:The Mathematical Association of America
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315:that are clopen is a Boolean algebra.
588:
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388:; the theorem states that there is a
339:). The isomorphism sends an element
266:(both closed and open). This is the
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194:consisting of all sets of the form
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515: – Functor in category theory
327:states that every Boolean algebra
38:it lacks sufficient corresponding
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439:The theorem is a special case of
268:topology of pointwise convergence
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311:, the collection of subsets of
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705:. Cambridge University Press.
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499:List of Boolean algebra topics
454:The proof requires either the
428:). In other words, there is a
325:Stone's representation theorem
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1:
723:A Course in Universal Algebra
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527: – Maximal proper filter
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548:Stone, Marshall H. (1936).
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460:Boolean prime ideal theorem
180:two-element Boolean algebra
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761:Theorems in lattice theory
277:For every Boolean algebra
673:; Givant, Steven (1998).
299:; such spaces are called
404:from a Boolean algebra
53:more precise citations.
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624:Doctor, H. P. (1964).
487:Representation theorem
449:partially ordered sets
381:is a Boolean algebra.
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358:{\displaystyle b\in B}
319:Representation theorem
258:. These sets are also
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170:, or equivalently the
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589:Dimov, G. D. (2012).
430:contravariant functor
408:to a Boolean algebra
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596:Publ. Math. Debrecen
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323:A simple version of
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182:. The topology on
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51:introducing
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664:References
162:) are the
95:isomorphic
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535:Citations
350:∈
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59:June 2015
701:(1982).
476:See also
394:category
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390:duality
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