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 ...