to PTJ

["Joyal, André" <joyal.andre@uqam.ca>]

Hi Peter,

>And, as someone (I forget who, but it may have been Mike Barr) pointed
>out long ago, one can (well, almost) define the variety of groups
>as the variety defined by a single binary operation satisfying a
>single equation; 1 < 3, but no sane group-theorist would do it
>that way.

I recall that Tarski is responsible for describing
the theory of groups with a single binary operation satisfying a
single equation. But dont have a reference with me.

A small point: the algebraic theory described by Tarski
admits the empty set as a model. Strictly speaking,
it is not equivalent to the theory of group. 

Best,
André



-------- Message d'origine--------
De: categories@mta.ca de la part de Prof. Peter Johnstone
Date: ven. 13/11/2009 16:49
À: Vaughan Pratt; categories@mta.ca
Objet : categories: Re: Lambek's lemma
 
Dear Vaughan,

Of course I agree with you that, logically, there is no point in
drawing a commutative square to prove that x = x. I also agree that
5 < 7. But I think there is still some point in drawing the second
square in A1.1.4, at least in pedagogical terms: until you've seen
(or at least visualized) the second square, it's hard for the mind
to accept the argument that says af = 1. (I feel strongly about this,
having spent two hours this afternoon in an examples class for the
students attending my first-year graduate course on category theory.)
And, as someone (I forget who, but it may have been Mike Barr) pointed
out long ago, one can (well, almost) define the variety of groups
as the variety defined by a single binary operation satisfying a
single equation; 1 < 3, but no sane group-theorist would do it
that way.

Peter

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