Re: Stone duality for generalized Boolean algebras

[Yoshihiro Maruyama <maruyama@i.h.kyoto-u.ac.jp>]

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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