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Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

Abstract

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.

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Correspondence to Nick Bezhanishvili.

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Bezhanishvili, G., Bezhanishvili, N., Sourabh, S. et al. Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality. Appl Categor Struct 25, 381–401 (2017). https://doi.org/10.1007/s10485-016-9434-2

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Keywords

  • Compact Hausdorff space
  • Stone space
  • Extremally disconnected space
  • Gleason cover
  • Boolean algebra
  • Complete Boolean algebra
  • Modal algebra
  • Proximity

Mathematics Subject Classification (2010)

  • 03B45
  • 54E05
  • 06E15
  • 54G05