Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (...

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Bibliographic Details
Published in:Applied categorical structures 2017-06, Vol.25 (3), p.381-401
Main Authors: Bezhanishvili, Guram, Bezhanishvili, Nick, Sourabh, Sumit, Venema, Yde
Format: Article
Language:English
Subjects:
Algebra
Boolean algebra
Convex and Discrete Geometry
Equivalence
Geometry
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Theory of Computation
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Summary:By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair ( X , R ), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X . Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.
ISSN:0927-2852
1572-9095
DOI:10.1007/s10485-016-9434-2