From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6482 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Stone duality for generalized Boolean algebras Date: Fri, 21 Jan 2011 14:19:12 +0100 Message-ID: Reply-To: Andrej Bauer NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1295640470 22265 80.91.229.12 (21 Jan 2011 20:07:50 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 21 Jan 2011 20:07:50 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Fri Jan 21 21:07:46 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PgNGs-0001OB-4u for gsmc-categories@m.gmane.org; Fri, 21 Jan 2011 21:07:46 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:54744) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PgNGU-0003kH-9a; Fri, 21 Jan 2011 16:07:22 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PgNGO-0006S2-HM for categories-list@mlist.mta.ca; Fri, 21 Jan 2011 16:07:17 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6482 Archived-At: 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]