From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2043 Path: news.gmane.org!not-for-mail From: Charles Wells Newsgroups: gmane.science.mathematics.categories Subject: Defining monoids Date: Thu, 06 Dec 2001 15:58:53 -0500 Message-ID: <5.1.0.14.2.20011206155739.0204cd50@mail.oberlin.net> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Trace: ger.gmane.org 1241018365 2180 80.91.229.2 (29 Apr 2009 15:19:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:19:25 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Dec 7 20:08:45 2001 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Dec 2001 20:08:45 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 3.33 #2) id 16CUsQ-0000up-00 for categories-list@mta.ca; Fri, 07 Dec 2001 19:57:58 -0400 X-Sender: cwells@mail.oberlin.net X-Mailer: QUALCOMM Windows Eudora Version 5.1 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 16 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:2043 Archived-At: In talking about defining monoids, Toby Bartels wrote: "We could also go straight to trees and define them as the basic operations, then requiring as axiom that grafting of trees produces the same result as composing the operations." This is the mu operation of the corresponding monad. Every single-sorted "idea" in the sense of the recent discussion generates a monad in sets with a mu like this. For each set S there is a set of possible computations TS, a mu:TTS to TS, and a "OneIdentity" operation in the sense of Mathematica that says the computation consisting of a single node results in that node; these subject to the monad laws. In other words, the phenomenon you noted is an instance of a general result. --Charles Wells Charles Wells, Emeritus Professor of Mathematics, Case Western Reserve University Affiliate Scholar, Oberlin College Send all mail to: 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 genealogical website: http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/ NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm