From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6514 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: categories with several compositions? Date: Wed, 02 Feb 2011 19:05:47 -0500 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1296693370 7409 80.91.229.12 (3 Feb 2011 00:36:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 3 Feb 2011 00:36:10 +0000 (UTC) To: John Stell , categories[at]mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Feb 03 01:36:05 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 1PknB5-0004hp-4Z for gsmc-categories@m.gmane.org; Thu, 03 Feb 2011 01:36:03 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:51828) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PknAx-0006Cp-Eo; Wed, 02 Feb 2011 20:35:55 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PknAp-0000Zi-3K for categories-list@mlist.mta.ca; Wed, 02 Feb 2011 20:35:47 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6514 Archived-At: On Wed, 02 Feb 2011 09:20:11 AM EST, John Stell asked: > 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 a single identity that works for both and where > (x *i y) *j z =3D x *i (y *j z) for all i,j ... What am I missing when I think I see, using y =3D 1 (an identity map, as suggested by the additional remarks Stell makes below), that x *j z =3D (x *i 1) *j z =3D x *i (1 *j z) =3D x *i z ? Cheers, -- Fred -- = > = > More generally, a kind of category with several compositions: > for each object y there is a set Dy and instead of the usual > = > C(x,y) x C(y,z) -> C(x,z) > = > we have Dy -> [C(x,y) x C(y,z), C(x,z)] > = > So you have a family of compositions at each object which "associate wi= th > each other" in the manner of the above equation, and where there is > a single identity for each object. > = > thanks > John Stell [For admin and other information see: http://www.mta.ca/~cat-dist/ ]