Re: Does commutativity imply associativity?
From: Dave Rusin (rusin_at_vesuvius.math.niu.edu)
Date: 08/28/04
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Date: 28 Aug 2004 01:59:35 GMT
In article <12f59340.0408271435.4abe582c@posting.google.com>,
Brian Smith <brianscsmith@yahoo.com> wrote:
>I know that there are mathematical objects which are associative but
>not commutative with respect to some binary operation. Matrix
>multiplication is an example, associative but not commutative.
>
>My question is: Is there some set of objects which is commutative but
>not associative with respect to some binary operation, or does
>commutativity imply associativity?
A classical example arises in Jordan algebras: you might for example
define a new produce "&" on the set of square matrices of a given
size by A & B = (AB + BA) , which is clearly commutative but not
associative (try e.g. A & ( A & B ) and ( A & A ) & B ).
dave
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