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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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637:"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
380:. This is a clopen set because of the choice of topology on
112:. 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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of homomorphisms into the two-element
Boolean algebra.
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690:. Dolciani Mathematical Expositions. Vol. 21.
602:"Some generalizations of the Stone Duality Theorem"
561:"The Theory of Representations of Boolean Algebras"
98:
Stone's representation theorem for
Boolean algebras
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253:
566:Transactions of the American Mathematical Society
454:, a more general framework for dualities between
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731:Burris, Stanley N.; Sankappanavar, H.P. (1981).
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395:Restating the theorem using the language of
376:to the set of all ultrafilters that contain
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318:). Conversely, given any topological space
18:Representation theorem for Boolean algebras
120:. Stone was led to it by his study of the
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254:{\displaystyle \{x\in S(B)\mid b\in x\},}
80:Learn how and when to remove this message
43:This article includes a list of general
692:The Mathematical Association of America
14:
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326:that are clopen is a Boolean algebra.
599:
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399:; the theorem states that there is a
350:). The isomorphism sends an element
277:(both closed and open). This is the
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205:consisting of all sets of the form
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526: – Functor in category theory
338:states that every Boolean algebra
49:it lacks sufficient corresponding
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450:The theorem is a special case of
279:topology of pointwise convergence
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322:, the collection of subsets of
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716:. Cambridge University Press.
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510:List of Boolean algebra topics
465:The proof requires either the
439:). In other words, there is a
336:Stone's representation theorem
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734:A Course in Universal Algebra
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538: – Maximal proper filter
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559:Stone, Marshall H. (1936).
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471:Boolean prime ideal theorem
191:two-element Boolean algebra
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772:Theorems in lattice theory
288:For every Boolean algebra
684:; Givant, Steven (1998).
310:; such spaces are called
415:from a Boolean algebra
64:more precise citations.
655:10.4153/CMB-1964-022-6
635:Doctor, H. P. (1964).
498:Representation theorem
460:partially ordered sets
392:is a Boolean algebra.
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369:{\displaystyle b\in B}
330:Representation theorem
269:. These sets are also
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441:contravariant functor
419:to a Boolean algebra
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607:Publ. Math. Debrecen
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334:A simple version of
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642:Canad. Math. Bull.
456:topological spaces
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193:. The topology on
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675:References
173:) are the
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664:124451802
546:Citations
361:∈
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234:∣
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70:June 2015
712:(1982).
487:See also
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302:compact
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