From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3496 Path: news.gmane.org!not-for-mail From: "mhebert" Newsgroups: gmane.science.mathematics.categories Subject: Re: Implicit algebraic operations Date: Mon, 27 Nov 2006 17:48:21 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 X-Trace: ger.gmane.org 1241019341 9007 80.91.229.2 (29 Apr 2009 15:35:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:35:41 +0000 (UTC) To: "categories" Original-X-From: rrosebru@mta.ca Tue Nov 28 09:37:41 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Nov 2006 09:37:41 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Gp33D-00070h-LA for categories-list@mta.ca; Tue, 28 Nov 2006 09:31:07 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 111 Xref: news.gmane.org gmane.science.mathematics.categories:3496 Archived-At: Hi everyone, Todd Wilson asks : > My question is this: Does a definitive treatment of this phenomenon [pa= rtial operations , ..., non-surjective epimorphisms,...] in > "algebraic" categories exist? Are there still some mysteries/open probl= ems? It seems to me that the problem of "characterizing the algebraic theories giving rise to varieties where all= the epis are surjective" (posed by Bill Lawvere in Some algebraic problems in the context..., LNM 61 (1968) ) is still essentially open. Anyone knows otherwise? (A "classical" version might be to find a - syntactic- condition on the = equations necessary and sufficient to have all epis surjective in its cat= egory of its models) Michel Hebert Fromcat-dist@mta.ca Tocategories@mta.ca Cc DateSun, 26 Nov 2006 22:16:59 -0800 Subjectcategories: Implicit algebraic operations > I was going through some of my old notes today and came across > investigations I had done several years ago on implicit operations in > Universal Algebra. These are definable partial operations on algebras > that are preserved by all homomorphisms. Here are two examples: > > (1) Pseudocomplements in distributive lattices. Given a <=3D b <=3D c i= n a > distributive lattice, there is at most one b' such that > > b /\ b' =3D a and b \/ b' =3D c. > > Because lattice homomorphisms preserve these inequalities and equations= , > the uniqueness of pseudocomplements implies that, when they exist, they= > are also preserved by homomorphisms. > > (2) Multiplicative inverses in monoids. Similarly, given an element m > in a monoid (M, *, 1), there is at most one element m' such that > > m * m' =3D 1 and m' * m =3D 1. > > It follows that inverses, when they exist, are also preserved by monoid= > homomorphisms. > > Now, the investigation of these partial operations gets one quickly int= o > non-surjective epimorphisms, dominions in the sense of Isbell, algebrai= c > elements in the sense of Bacsich, implicit partial operations in the > sense of Hebert, and other topics. Some of the references that I know > about are listed below. > > My question is this: Does a definitive treatment of this phenomenon in > "algebraic" categories exist? Are there still some mysteries/open probl= ems? > > > REFERENCES > > PD Bacsich, "Defining algebraic elements", JSL 38:1 (Mar 1973), 93-101.= > > PD Bacsich, "An epi-reflector for universal theories", Canad. Math. > Bull. 16:2 (1973), 167-171. > > PD Bacsich, "Model theory of epimorphisms", Canad. Math. Bull. 17:4 > (1974), 471-477. > > JR Isbell, "Epimorphisms and dominions", Proc. of the Conference on > Categorical Algebra, La Jolla, Lange and Springer, Berlin 1966, 232-246= . > > JM Howie and JR Isbell, "Epimorphisms and dominions, II", J. Algebra, 6= > (1967), 7-21. > > JR Isbell, "Epimorphisms and Dominions, III", Amer. J Math. 90:4 (Oct > 1968), 1025-1030. > > M Hebert, "Sur les operations partielles implicites et leur relation > avec la surjectivite des epimorphismes", Can. J. Math. 45:3 (1993), 554= -575. > > M Hebert, "On generation and implicit partial operations in locally > presentable categories", Appl. Cat. Struct. 6:4 (Dec 1998), 473-488. > > -- > Todd Wilson A smile is not an individual > Computer Science Department product; it is a co-product. > California State University, Fresno -- Thich Nhat Hanh > >