From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3157 Path: news.gmane.org!not-for-mail From: wlawvere@buffalo.edu Newsgroups: gmane.science.mathematics.categories Subject: Re: Progressive or linear or ... monoids? Date: Sat, 25 Mar 2006 08:46:14 -0500 Message-ID: <1143294374.442549a6154a9@mail2.buffalo.edu> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241019126 7483 80.91.229.2 (29 Apr 2009 15:32:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:06 +0000 (UTC) To: categories list Original-X-From: rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNRwr-0004sP-47 for categories-list@mta.ca; Sun, 26 Mar 2006 05:54:13 -0400 In-Reply-To: X-Mailer: University at Buffalo WebMail Cyrusoft SilkyMail v1.1.11 X-Originating-IP: 128.205.248.196 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 103 Original-Lines: 52 Xref: news.gmane.org gmane.science.mathematics.categories:3157 Archived-At: Dear Vaughan A related question, originating with the problem of the quality of the truth value space in M- sets, led me to the discovery of an equational class that some years later, Peter Freyd rediscovered from an entirely different point of view. What is the Structure of the union of the "variety" groups with the variety of commutative monoids ? The motivation was the common feature of the Heyting algebra of right ideals of a monoid in either class, and the partial answer is in my "Taking categories seriously", reprinted in TAC. Not only do we have to think about the meaning of "union" and about the analogy with closed subschemes (probably the source of the term "variety") but also about the fact that the inclusion of groups in monoids is more "open" than closed and is certainly not a (sub) variety even though both categories are algebraic. Again analogously, the Structure 2-functor, adjoint to Semantics does not carry full inclusions to surjective interpretations, Thus in particular in my example like others, the algebraic theory that results has more operations, not only more equations. It may be true in your case as well. Bill Quoting Vaughan Pratt > 1. Is the quasivariety of monoids generated by the groups and the > free > monoids finitely based? > > That is, is there a finite set of universal Horn formulas entailing > the > common universal Horn theory of groups and free monoids? > > In other words, what do groups and free monoids have in common, > besides > being monoids? > > Apart from the (equational) axioms for monoids, the only members of > that > theory I can think of are xy=x -> y=1 and yx=x -> y=1. > > 2. How different is the abelian case? More or fewer axioms? > > Vaughan Pratt > > > >