From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: Multiplicatively closed "algebra"
Date: 2 Sep 2000 21:21:18 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <8onfrh$2eh$1@mortar.ucr.edu>,
Toby Bartels <toby@math-cl-n03.math.ucr.edu> wrote:

>Robert Hill wrote:

>>MCKAY john wrote:

>>>What is the correct name for an "algebra closed only under multiplication"?

>>I've seen "quasigroup" used for a set closed under a single binary operation
>>of which nothing more is assumed, but would not be surprised to learn that
>>the word has also been used with other meanings, or that other terms have
>>been used with this meaning.

>I've seen "groupoid", but this is falling out of favour,
>as the other meaning for "groupoid"[*] gains ground.

Yes, these terms are a bit dangerous, since some have multiple
definitions.  I have never seen "quasigroup" used in the sense Robert
Hill describes.  Here is what Eric Weisstein's Encyclopedia of
Mathematics says - just to give one viewpoint on these matters:

1) a "groupoid" is a set with a binary operation, OR
   a "groupoid" is a category with all morphisms invertible.

2) a "semigroup" is a set with a binary associative operation.

3) a "monoid" is a semigroup with an element e such that xe = x
and ex = x for all x.

3) a "quasigroup" is a set with a binary operation such that for all
a, b, there exist unique x, y such that ax = b and ya = b.

4) a "loop" is a quasigroup with an element e such that xe = x and
ex = x for all x.

5) a "group" is a monoid for which every element has a left and
right inverse, or equivalently, an associative loop.

To illustate some of the more esoteric of these concepts:

A) The octonions of norm 1 form a loop but not a group.  Has anyone
seriously tried doing gauge theory with a gauge loop instead of a
gauge group?  If so, the octonions of norm one (which form a 7-sphere)
would be the obvious example to try.  I believe they also form a Joyce
manifold, i.e. a Riemannian manifold with G2 holonomy.  If so, they are
also good for compactifying 11-dimensional supergravity!

B) Any Latin square gives a quasigroup.  A "Latin square" is an n x n
matrix of numbers from 1 to n, where each number appears exactly once
in each row and each column.  They play an important role in the
design of certain experiments.

>Bourbaki, I'm told, says "magma", which seems pretty good,
>on the grounds that it's unlikely to be used for anything else.

Yes, I like "magma" for a set with a binary operation.  I thus go
along with all of Eric Weisstein's definitions except that I save
"groupoid" to mean a category with all morphisms invertible, using
"magma" for his other sense of "groupoid".

While we're at it, I wonder how many people know the significance
of this sequence:

1, 2, 12, 576, 161280, 812851200, 61479419904000,
108776032459082956800, 5524751496156892842531225600,
9982437658213039871725064756920320000, ....

and which person named McKay has written about this sequence.  NOTE:
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