From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3418 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Linear--structure or property? Date: Mon, 04 Sep 2006 00:14:11 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019292 8650 80.91.229.2 (29 Apr 2009 15:34:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:52 +0000 (UTC) To: "Categories list" Original-X-From: rrosebru@mta.ca Mon Sep 4 09:32:37 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 04 Sep 2006 09:32:37 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GKDZz-0000I4-Jr for categories-list@mta.ca; Mon, 04 Sep 2006 09:29:31 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 68 Xref: news.gmane.org gmane.science.mathematics.categories:3418 Archived-At: Three items with which to follow up on my earlier note, > For instance, take the additive group of 2x2 matrices with > integer entries (or entries from any semiring) and notice that, > apart from the usual matrix multiplication, there is also > the sophomoric, or pointwise, multiplication ...: 1. In one of the poster sessions at the recent Madrid ICM2006, Camarero, Etayo, Rovira, and Santamaria remind us that the ordinary real plane R^2 (with usual vector addition) admits at least > three distinguished real algebras ... as follows: the set > {a+bi: a, b {/element} R} with i^2 =3D -1, +1, 0, i.e., the > complex, double, and dual numbers. (ICM2006 Abstracts, p. 42) 2. Yefim Katsov (in a telephone conversation) has pointed out that in any reasonable lattice-ordered (semi-)group, where a + (b ^ c) =3D (a+b) ^ (a+c) and a + (b v c) =3D (a+b) v (a+c), one gets two different (semi-)ring structures by using: = as product, the (semi-)group composition + ; and = as sum, in one case the lattice meet ^ , alternatively, the join v . 3. A many-objects version can be concocted from example 2 above by stirring it up with a variant form of Lawvere's observations about metric spaces being categories enriched over an appropriately structured closed monoidal version of the poset R+ of nonnegative real numbers ( order relation > , tensor product + , unit 0 , internal hom the positive part of b-a ). = In detail, if X is a metric space with metric d, consider the ordinary category _X_ whose objects are the points of X while its homsets _X_(p, q) are given by the principal filter (in (R, /=3D d(p, q) } . Composition _X_(p, q) x _X_(q, r) can clearly be given by sending (x, y) to x+y : for identity map is always the number 0, and whenever x >/=3D d(p, q) and y >/=3D d(q, r) we must also have x+y >/=3D d(p, q) + d(q, r) >/=3D d(p, r) . Thus, arithmetic addition provides a composition rule for _X_ , and both real sup and real inf can serve as commutative semigroup structures (across which composition distributes) on the homsets. Of course no one will claim _X_ has any finite (co)products; but, anyway, here any enrichment of _X_ over semigroups is clearly an added item of structure, and not a property of _X_ . -- Fred (and pardon, please, the crude ASCII/TeX symbology) PS: Katsov has also pointed out that a marvelous little New Yorker piece of Fields Medal gossip, turning around Yau, Perelman, Hamilton, and the Poincare conjecture, can be found on the web (for those who don't take the New Yorker, or even those who do) at: http://www.newyorker.com/printables/fact/060828fa_fact2 . -- F.