Re: Does commutativity imply associativity?
From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 08/27/04
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Date: Fri, 27 Aug 2004 23:00:01 +0000 (UTC)
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?
You can easily construct one. Take the following operation on the set
of positive integers:
a*b = max{a,b} + 10.
The operation is clearly commutative. But
(25*17)*10 = (35)*10 = 45
25*(17*10) = 25*(27) = 37.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@math.berkeley.edu
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