Dear Andrej, The following paper seems to provide some details of a duality between generalized Boolean algebras and locally compact zero-dimensional Hausdorff spaces: H. P. Doctor, The categories of Boolean lattices, Boolean rings, and Boolean spaces, Canadian Mathematical Bulletin 7 (1964) 245-252. It would be better to be careful of the morphism part of the duality in the paper, in which continuous proper maps of locally compact zero-dimensional Hausdorff spaces correspond to "proper" homomorphisms of generalized Boolean algebras (and, as you noted, do not correspond to all homomorphisms). Since the category of Boolean algebras is a full subcategory of GBA and their proper homomorphisms, the duality in the above paper is a generalization of Stone duality between Boolean algebras and compact zero-dimensional Hausdorff spaces. As you suggested, another way would be to extend morphisms of spaces (if we place emphasis on algebras rather than spaces). I wish this would be useful for you. (Sorry if I misunderstand anything.) With best regards, Yoshihiro ********************************************************** Yoshihiro Maruyama Department of Humanistic Informatics Kyoto University E-mail: maruyama@i.h.kyoto-u.ac.jp Webpage: http://researchmap.jp/ymaruyama/ ********************************************************** 2011/1/21 Andrej Bauer <andrej.bauer@andrej.com>:
The well known Stone duality says that there is an equivalence between Boolean algebras (BA) and the opposite of Stone spaces and continuous maps. Here a Stone space is a Hausdorff zero-dimensional compact space. Furthermore, Boolean algebras correspond to Boolean rings with unit.
How exactly does this extend to generalized Boolean algebras? A generalized Boolean algebra (GBA) is an algebra with 0, binary meet, binary join, and relative complement in which meets distribute over joins. Equivalently it is a Boolean ring (possibly without a unit). I have seen it stated that the dual to these are (the opposite of) locally compact zero-dimensional Hausdorff spaces and proper maps, e.g., it is stated in Benjamin Steinberg: "A groupoid approach to discrete inverse semigroup algebras", Advances in Mathematics 223 (2010) 689-727.
Another source to look at is Givant & Halmos "Introduction to Boolean algebras", but there this material is covered in exercises and the duality is stated separately for objects and for morphisms, and I can't find an exercise that treats the morphisms, so I wouldn't count that as a reliable reference. Stone's original work does not seem to speak about morphisms very clearly (to me).
Unless I am missing something very obvious, it cannot be the case that GBA's correspond to locally compact 0-dimensional Hausdorff spaces and proper maps, for the following reason.
The space which corresponds to the GBA 2 = {0,1} is the singleton. The space which corresponds to the four-element GBA 2 x 2 is the two-point discrete space 2. There are _four_ GBA homomorphisms from 2 to 2 x 2 (because a GBA homomorphism preserves 0 but it need not preserve 1), but there is only one continuous map from 2 to 1. Or to put it another way, there are _four_ ring homomorphisms from Z_2 to Z_2 x Z_2 (because they need not preserve 1), but there is only one continuous map from 2 to 1. So, either the spectrum of a GBA is not what I think it is (namely maximal ideals), or we should be taking a more liberal notion of maps on the topological side. For example, there are _four_ partial maps from 2 to 1.
If someone knows of a reliable reference, one would be much appreciated. I won't object to a direct proof of duality either.
With kind regards,
Andrej
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