From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/662 Path: news.gmane.org!not-for-mail From: "Uday S. Reddy" Newsgroups: gmane.science.mathematics.categories Subject: Quantifiers for monoids Date: Sat, 28 Feb 1998 23:24:32 -0600 (CST) Message-ID: <199803010524.XAA25141@reddy.cs.uiuc.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017106 26779 80.91.229.2 (29 Apr 2009 14:58:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:58:26 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Sun Mar 1 11:53:22 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA02389 for categories-list; Sun, 1 Mar 1998 10:55:44 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:662 Archived-At: In studying Algol-like languages, I repeatedly run into operators that have an interesting structure. I am wondering if such operators are studied somewhere. Consider a monoid in a CCC. The operations of interest are natural transformations E_A : [A => M] -> M that satisfy the following equations (in the internal language of the CCC): E_A(\lambda x. 1) = 1 E_A(\lambda x. a * g`x) = a * E_A(g) E_A(\lambda x. g`x * a) = E_A(g) * a E_A(\lambda x. E_B(\lambda y. h`x`y)) = E_B(\lambda y. E_A(\lambda x. h`x`y)) These operators "feel" like existential quantifiers. In fact, if M is a subobject classifier with the monoid structure ofh conjunction, then the existential quantifier E satisfies all of these equations (though it is not a natural transformation). In the applications I am interested in, M is a type of commands, with * as sequential composition and 1 as the null action. An example of E is a local variable declaration. Is there some algebra or theory related to these kinds of operators? Cheers, Uday Reddy