Date: Wed, 13 Dec 95 09:31:02 CST
From: rusin (Dave Rusin)
To: atrojan@yorku.ca
Subject: Re: Looped
Newsgroups: sci.math
In article <4am43p$ou7@sunburst.ccs.yorku.ca> you write:
> I am looking for the answer to the following questions:
> 1. What is a Moufang Loop?
I'm short of details, but:
A loop is a object which satisfies some of the axioms of a group.
More precisely, it's a set L with a binary operation on it and a
distinguished element e such that
(a) for all x in L, xe=ex=x
(b) for all a, b in L there is a unique x with ax=b and
there is a unique y with ya=b
Notice that if associativity were assumed (which it isn't), L would be
a group.
A Moufang loop is a loop which satisfies some extra axiom(s) in
the form of an equation to hold for all elements of the group. I can't
recall if it's supposed to be a weaker form of associativity (e.g.
x(yy)x=(xy)(yx) for all x and y) or something rather distinct from
associativity (e.g. (xy)z=(xz)(yz) ).
> 2. What good is one?
That's always a loaded question to ask about a mathematical construct.
I think the answer is that Moufang was considering projective planes
(sets with subsets called "lines" and "points", satisfying certain
incidence axioms). These tend to look like coordinate systems over rings,
but not until you add some reasonable geometric assumptions. I'm
guessing that Moufang looked to see the algebraic consequences of
insisting that the projective plane have some reasonable geometric
property (in his case, probably something like "there exist translations
along every line") and found that this implied some algebraic object
formed a Moufang loop.
I don't think this material is trendy today, but it was pretty well
studied through the 60s and 70s.
dave
==============================================================================
Correction 2002/01/12:
Ruth Moufang was not looking at "his" case. For more information visit
http://www.agnesscott.edu/lriddle/women/moufang.htm