Re: Stone duality for generalized Boolean algebras

["George Janelidze" <janelg@telkomsa.net>]

The answer is trivial: The category of Boolean rings without 1 (which means
"possibly without 1" of course) is equivalent to BA/2, which, by Stone
duality, is dually equivalent to the category of pointed Stone spaces.

However thinking of "partial maps" was not too bad since, say, the category
of pointed sets is equivalent to the category of sets with partial maps as
morphisms.

George Janelidze

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From: "Andrej Bauer" <andrej.bauer@andrej.com>
Sent: Friday, January 21, 2011 3:19 PM
To: "categories list" <categories@mta.ca>
Subject: categories: Stone duality for generalized Boolean algebras

> 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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