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A Refinement of Stone Duality to Skew Boolean Algebras

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A Refinement of Stone Duality to Skew Boolean Algebras A refinement of Stone duality to skew Boolean algebras Ganna Kudryavtseva Abstract. We establish two theorems which refine the classical Stone duality be- tween generalized Boolean algebras and locally compact Boolean spaces. In the first theorem we prove that the category of left-handed skew Boolean algebras whose mor- phisms are proper skew Boolean algebra homomorphisms is equivalent to the category of ´etalespaces over locally compact Boolean spaces whose morphisms are ´etalespace cohomomorphisms over continuous proper maps. In the second 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 ´etalespaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective ´etalespace cohomomorphisms over continuous proper maps. 1. Introduction The aim of the present paper is to refine the classical Stone duality [20, 7] between the categories of generalized Boolean algebras and locally compact Boolean spaces to two equivalences between the categories of left-handed skew Boolean algebras and ´etalespaces over locally compact Boolean spaces. Skew Boolean algebras are non-commutative generalizations of Boolean al- gebras, all of whose elements are idempotents with respect to both ^ and _. In the present paper we consider left-handed skew Boolean algebras whose structure is as follows. For a left-handed skew Boolean algebra S there is a congruence D on S such that S=D is a generalized Boolean algebra (in fact, a maximal generalized Boolean algebra image of S), and the D-classes are flat left-handed subalgebras of S: for any x; y such that x D y we have x ^ y = x and x _ y = y. Our first main result is the following theorem: Theorem 1.1. The category Skew of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category Etale of ´etalespaces over locally compact Boolean spaces whose morphisms are ´etalespace cohomomorphisms over continuous proper maps. Let us briefly explain the idea, that underlies Theorem 1.1, how to connect skew Boolean algebras and ´etalespaces. Given a left-handed skew Boolean Presented by . Received . ; accepted in final form . 2010 Mathematics Subject Classification: Primary: 06E75; Secondary: 06E15, 03G05, 18B30, 18F20, 54B40. Key words and phrases: skew Boolean algebras, generalized Boolean algebras, locally compact Boolean spaces, Stone duality, ´etalespaces, cohomomorphisms of ´etalespaces. 2 Ganna Kudryavtseva Algebra univers. algebra S we introduce the notion of filters of S as subsets of S that are closed upwards and are closed with respect to the operation ^. Then we introduce prime filters of S as minimal filters of S whose image in S=D is a prime filter of S=D. We consider two topological spaces: the spectrum S? of S that is the space of prime filters of S, and the spectrum (S=D)? of S=D, that is the space of prime filters of S=D. Prime filters of S turn out to be related to prime filters of S=D in a way that gives rise to a surjective local homeomorphism π : S? ! (S=D)?. The triple (S?; π; (S=D)?) is therefore an ´etalespace over (S=D)?. Conversely, given an ´etalespace (E; π; X), where X is a locally compact Boolean space, one constructs a left-handed skew Boolean algebra E? as follows: the elements of E? are sections of E over compact clopen sets, and the idempotent binary operations ^ and _ are the operations of quasi- intersection and quasi-union of sections, that generalize the usual intersection and union operations on sets. On the morphism front, the topological counterparts of homomorphisms of skew Boolean algebras are ´etalespace cohomomorphisms, that generalize opposite maps to continuous maps between topological spaces. In the commu- tative Stone duality [7], proper homomorphisms of generalized Boolean alge- bras correspond to continuous proper maps on topological side. We refine this by establishing a correspondence between proper skew Boolean algebra ho- momorphisms and cohomomorphisms of ´etalespaces over continuous proper maps. On one hand, an ´etalespace f-cohomomorphism E G, with f being proper, maps sections over compact clopen sets of E to such sections of G and preserves the operations of quasi-union and quasi-intersection. Therefore, it gives rise to a proper skew Boolean algebra homomorphism E? ! G?. On the other hand, given a proper skew Boolean algebra homomorphism S ! T , for any prime filter F of T the inverse image of F is a union of prime filters of S that are above the same prime filter of S=D. This gives rise to an ´etalespace ? ? f-cohomomorphism S T with f being proper. Our second main result is a generalization of Stone duality to left-handed skew Boolean algebras with finite intersections, or skew Boolean \-algebras. A skew Boolean algebra S is said to have finite intersections, if for any a; b 2 S there exists the greatest lower bound a \ b of a and b, called the intersection of a and b, with respect to the natural partial order. A skew Boolean \-algebra is a skew Boolean algebra S with finite intersections whose signature is enriched by the binary operation of intersection. Theorem 1.2. The category SkewInt of left-handed skew Boolean \-algebras whose morphisms are proper skew Boolean \-algebra homomorphisms is equiv- alent to the category EtaleEq of ´etalespaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective ´etalespace cohomomorphisms over continuous proper maps. In fact, Theorem 1.2 is a consequence of Theorem 1.1 and the two fol- lowing observations. The first observation is that topological counterparts of Vol. 00, XX A refinement of Stone duality to skew Boolean algebras 3 left-handed skew Boolean \-algebras are ´etalespaces (E; π; X) satisfying a restriction that is called the equalizer condition. The latter condition is equiv- alent to E being a Hausdorff space. The second observation is that there is a correspondence between proper skew Boolean \-algebra homomorphisms and ´etalespace f-cohomomorphisms, with f proper, all of whose components are one-to-one. A different view of Stone duality for skew Boolean \-algebras has recently appeared in [2]. Another idea to generalize Boolean algebras is to consider inverse semi- groups whose idempotents form a Boolean algebra. This approach is realized in recent papers [11, 12], where a duality between Boolean inverse semigroups and Boolean groupoids is established, see also a generalization to a point-free setting [13]. The outline of this paper is as follows. In Section 2 we review the classical Stone duality. Then in Sections 3 and 4 we collect preliminaries on skew Boolean algebras and ´etalespaces. Further, in Sections 5 and 6 we construct the functors SB : Etale ! Skew and ES : Skew ! Etale. In Sections 7 and 8 we give proofs of Theorems 1.1 and 1.2. We also explain how to transform the ´etalespace representation of a skew Boolean algebra into a subdirect product representation. The latter, in the case of skew Boolean \-algebras whose maximal lattice images are Boolean algebras, is isomorphic to the Boolean product representation of [17, 4.10]. Finally, we discuss the connection of our duality for skew Boolean \-algebras with the generalization of Stone duality for varieties generated by quasi-primal algebras given in [10]. Acknowledgements The author thanks Andrej Bauer, Karin Cvetko-Vah, Jeff Egger and Alex Simpson for their comments, and the anonymous referee for the suggestions on improvement of the introduction and notation. 2. Classical Stone duality Here we briefly review the classical Stone duality [20, 7] between generalized Boolean algebras and locally compact Boolean spaces. A detailed exposition of the duality between Boolean algebras and Boolean spaces can be found in most of the textbooks on Boolean algebras, e.g., in [5, 8, 9]. An exposition of the discrete case of the latter duality with an excellent insight to the categorical background can be found in [1]. Let GBA be the category, whose objects are generalized Boolean algebras (i.e, relatively complemented distributive lattices with a zero) and whose morphisms are proper homomorphisms of generalized Boolean algebras. Recall that a homomorphism f : B1 ! B2 of generalized Boolean algebras is called proper [7], provided that for any c 2 B2 there exists b 2 B1, such that f(b) ≥ c. By BA we denote the category of Boolean algebras 4 Ganna Kudryavtseva Algebra univers. and homomorphisms of Boolean algebras. A topological space X is called locally compact [8, p. 335] if for every point x, there is a compact set K whose interior contains x. A locally compact Hausdorff space in which the clopen sets form a base for the topology is called locally compact Boolean space. In such a space compact clopen sets form a base, and every compact open set is clopen. A continuous map X ! Y of topological spaces is called proper if the inverse image of each compact set is compact. Denote by LCBS the category, whose objects are locally compact Boolean spaces and whose morphisms are continuous proper maps between such spaces. If a locally compact Boolean space is compact then it is called Boolean space. Denote by BS the category whose objects are Boolean spaces and whose morphisms are continuous maps. By 2 = f0; 1g we denote the two-element Boolean algebra. For a generalized Boolean algebra B denote by B? the set of all prime filters on B, that is the set of all inverse images of 1 under nonzero homomorphisms B ! 2. For A 2 B set M(A) = fF 2 B? : A 2 F g. The sets M(A) generate a topology on B?, that turns B? into a locally compact Boolean space.
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