From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6515 Path: news.gmane.org!not-for-mail From: N.Bowler@dpmms.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: RE: categories with several compositions? Date: 03 Feb 2011 11:36:58 +0000 Message-ID: References: Reply-To: N.Bowler@dpmms.cam.ac.uk NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 X-Trace: dough.gmane.org 1296833045 3661 80.91.229.12 (4 Feb 2011 15:24:05 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 4 Feb 2011 15:24:05 +0000 (UTC) Cc: "'categories@mta.ca'" To: John Stell Original-X-From: majordomo@mlist.mta.ca Fri Feb 04 16:24:00 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 1PlNVw-0008Ms-4o for gsmc-categories@m.gmane.org; Fri, 04 Feb 2011 16:24:00 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34158) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PlNVY-0006sn-BM; Fri, 04 Feb 2011 11:23:36 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PlNVT-0008Cs-5t for categories-list@mlist.mta.ca; Fri, 04 Feb 2011 11:23:31 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6515 Archived-At: > Can anyone tell me whether these structures have been studied anywhere? > > A kind of generalized monoid with two or more compositions *1, *2, etc > with [each composition having its own identity] and where > (x *i y) *j z = x *i (y *j z) for all i,j We can certainly relate these structures to things we already understand reasonably well (though I can't see how to say `they are just wombats'). First of all, here's a way to generate lots of examples. Pick any monoid M, and let (e_i | i is in I) be any family of invertible elements of M. For each i in I, define a new operation *i on M by a *i b = a * (e_i^{-1}) * b. M is a monoid under each *i, with identities the s_i, and (x *i y) *j z = x *i (y *j z) for all i,j. In fact, all examples arise in this way. Pick one of the operations, which we will treat differently from the others: call it * and call its identity e. So the structure is a monoid with respect to * and e. I'll call this monoid M. Now pick some other operation *i and let e_i be the identity for *i. e_i * (e *i e) = (e_i * e) *i e = e_i *i e = e and similarly (e *i e) * e_i = e, so e_i is an invertible element of M with inverse (e *i e). Now note that for any a and b, a *i b = (a * e) *i (e * b) = a * (e *i e) * b = a * (e_i^{-1}) * b so that each *i arises as above. On the basis of this analysis, it looks like the structures you asked about bear the same sort of relation to monoids with a designated family of invertible elements that torsors do to groups. Nathan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]