From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2253 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: equivalent varieties Date: Sat, 26 Apr 2003 22:47:14 +0100 (BST) Message-ID: References: <200304251826.h3PIQtRn001255@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018527 3262 80.91.229.2 (29 Apr 2009 15:22:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:07 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Apr 27 15:38:23 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Apr 2003 15:38:23 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 199qyB-0003TR-00 for categories-list@mta.ca; Sun, 27 Apr 2003 15:33:47 -0300 In-Reply-To: <200304251826.h3PIQtRn001255@saul.cis.upenn.edu> X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *199XVr-0001II-3D*V.UUzKy7mOA* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 47 Xref: news.gmane.org gmane.science.mathematics.categories:2253 Archived-At: On Fri, 25 Apr 2003, Peter Freyd wrote: > Varieties of algebras when viewed as categories can be unexpectedly > equivalent. For a reason explained at the end, I was looking at > varieties of unital rings satisfying the equations p = 0 and > x^p = x, one such variety for each prime integer p. > > The equivalence type of these categories is independent of p. The > easiest way of establishing that is to show that each is equivalent to > the category of Boolean algebras (a well-known fact when p = 2) and > all the equivalences can by established by just one functor. Given a > unital ring, R, define B(R) to be the boolean algebra of its central > idempotents where the meet of a and b is ab and the join is > a + b - ab. Then the restriction of B to the p'th variety described > above is always an equivalence of categories. > > The fastidious will note (one would certainly hope) that B is not a > functor in general (homomorphisms don't preserve centrality). But in a > ring "without nilpotents" (that is, in which x^2 = 0 implies x = 0) > all idempotents are central. The equations x^p = x, of course, imply > the absence of nilpotents. > > (Given p the inverse functor to B can be described as follows: for > a Boolean algebra C consider the set of "p-labeled partitions of > unity", that is, the set of functions f:Z_p -> C whose values are > pairwise disjoint and have unity as their join. Given two such, f and > g, define their sum by setting (f+g)i to be the join of the set > { fj ^ gk | j+k = i } and their product by setting (fg)i to be the > join of { fj ^ gk | jk = i }.) > The equivalence of these varieties for all p is well known. It's best understood by seeing that they are all dual to the category of Stone spaces: given a Stone space, the ring of continuous Z_p-valued functions on it (where Z_p is given the discrete topology) is a ring satisfying p1=0 and x^p=x; conversely, given such a ring, its prime (=maximal) ideal spectrum is a Stone space. Not having my copy of "Stone Spaces" to hand as I write this, I can't remember whether this fact was in the book. But it certainly should have been. Peter Johnstone