Stone duality for generalized Boolean algebras

[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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