Abstract
We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean \({\bigcap}\)-algebras whose morphisms are proper skew Boolean \({\bigcap}\)-algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.
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References
Awodey S.: Category Theory. Oxford Logic Guides, vol 49. Oxford University Press, New York (2006)
Bauer, A., Cvetko-Vah, K.: Stone duality for skew Boolean algebras with intersections. Houston J. Math., in press
Bignall R.J., Leech J.E.: Skew Boolean algebras and discriminator varieties. Algebra Universalis 33, 387–398 (1995)
Bredon, G.E.: Sheaf Theory, 2nd edn. Graduate Texts in Mathematics, vol. 170. Springer, New York (1997)
Burris S., Sankappanavar H.P.: A Course in Universal Algebra.Graduate Texts in Mathematics, vol 78. Springer, New York (1981)
Cornish W.H.: Boolean skew algebras. Acta Math. Acad. Sci. Hungar. 36, 281–291 (1980)
Doctor H.P.: The categories of Boolean lattices, Boolean rings and Boolean spaces. Canad. Math. Bull. 7, 245–252 (1964)
Givant S., Halmos P.: Introduction to Boolean Algebras. Springer, New York (2009)
Johnstone P.T.: Stone Spaces. Cambridge University Press, Cambridge (1986)
Keimel K., Werner H.: Stone duality for varieties generated by quasi-primal algebras. Mem. Amer. Math. Soc. 148, 59–85 (1974)
Lawson M.V.: A non-commutative generalization of Stone duality. J. Aust. Math. Soc. 88, 385–404 (2010)
Lawson, M.V.: Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras. arXiv1104.1054v1
Lawson, M.V., Lenz, D.H.: Pseudogroups and their étale groupoids. arXiv:1107.5511v2
Leech J.: Skew lattices in rings. Algebra Universalis 26, 48–72 (1989)
Leech J.: Skew Boolean algebras. Algebra Universalis 27, 497–506 (1990)
Leech J.: Normal skew lattices. Semigroup Forum 44, 1–8 (1992)
Leech J.: Recent developments in the theory of skew lattices. Semigroup Forum 52, 7–24 (1996)
Mac Lane S., Moerdijk I.: Sheaves in Geometry and Logic. Springer, New York (1994)
Steinberg B.: A groupoid approach to discrete inverse semigroup algebras. Adv. Math. 223, 689–727 (2010)
Stone M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41, 375–481 (1937)
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Communicated by M. Jackson.
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Kudryavtseva, G. A refinement of Stone duality to skew Boolean algebras. Algebra Univers. 67, 397–416 (2012). https://doi.org/10.1007/s00012-012-0192-1
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DOI: https://doi.org/10.1007/s00012-012-0192-1