From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1813 Path: news.gmane.org!not-for-mail From: Charles Wells Newsgroups: gmane.science.mathematics.categories Subject: Why binary products are ordered Date: Mon, 29 Jan 2001 10:18:52 -0800 Message-ID: <4.1.20010129101330.00ad8560@mail.oberlin.net> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241018121 603 80.91.229.2 (29 Apr 2009 15:15:21 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:15:21 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jan 29 18:08:57 2001 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f0TLBLU03429 for categories-list; Mon, 29 Jan 2001 17:11:21 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: cwells@mail.oberlin.net X-Mailer: QUALCOMM Windows Eudora Pro Version 4.1 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 44 Original-Lines: 24 Xref: news.gmane.org gmane.science.mathematics.categories:1813 Archived-At: One might say that the ordering of binary products, with a first projection and a second projection, is spurious but inevitable. The two components of a binary product must be distinguished, as Colin McLarty explained, but they must be allowed to be isomorphic. The usual way we handle a situation like this in mathematics is to index them, in this case by a two-element set. One could use {red,blue}. As far as I know, in Western culture, this set has no canonical ordering, but nevertheless one knows that there is a redth component and a blueth component and they might or might not be different objects. However, in practice the index set is {0,1}, {1,2} or {x,y} (the latter in analytic geometry). All of these are canonically totally ordered in our culture, so inevitably binary products do have an order in practice. Charles Wells, 105 South Cedar St., Oberlin, Ohio 44074, USA. email: charles@freude.com. home phone: 440 774 1926. professional website: http://www.cwru.edu/artsci/math/wells/home.html personal website: http://www.oberlin.net/~cwells/index.html NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm