Re: Linear--structure or property?

["Fred E.J. Linton" <fejlinton@usa.net>]

George Janelidze and others answer affirmatively the question

> Could you have two (semi)ring structures on the same set 
> with the same associative multiplication?

(attributed to Mike Barr) without ever noticing that the
related question, of having two (semi)ring structures 
on the same set, with the same addition, also has answer YES.

For instance, take the additive group of 2x2 matrices with
integer entries (or entries from any semiring) and notice that,
apart from the usual matrix multiplication, there is also
the sophomoric, or pointwise, multiplication (so called since
it is generally only sophomores in the first week of their
first linear algebra course who, following the analogous
pointwise definition of matrix addition, would wish to multiply 
two matrices by multiplying their corresponding entries).

Not quite sure though how this impacts the situation with
more than one object.

-- Fred

------ Original Message ------
Received: Fri, 11 Aug 2006 01:06:56 PM EDT
From: "George Janelidze" <janelg@telkomsa.net>
To: "Categories list" <categories@mta.ca>
Subject: categories: Re: Linear--structure or property?

> Dear Steve,
> 
> It is true that constructing such examples with more than one object is ...