From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6489 Path: news.gmane.org!not-for-mail From: Yoshihiro Maruyama Newsgroups: gmane.science.mathematics.categories Subject: Re: Stone duality for generalized Boolean algebras Date: Sun, 23 Jan 2011 13:06:17 +0900 Message-ID: References: Reply-To: Yoshihiro Maruyama NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1295783981 28596 80.91.229.12 (23 Jan 2011 11:59:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 23 Jan 2011 11:59:41 +0000 (UTC) Cc: categories list To: Andrej Bauer Original-X-From: majordomo@mlist.mta.ca Sun Jan 23 12:59:36 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 1PgybX-0003b6-Uf for gsmc-categories@m.gmane.org; Sun, 23 Jan 2011 12:59:36 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:49415) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PgybP-0007Qt-4g; Sun, 23 Jan 2011 07:59:27 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PgybL-0004FH-5k for categories-list@mlist.mta.ca; Sun, 23 Jan 2011 07:59:23 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6489 Archived-At: 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 : > 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/ ]