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Topic: A natural example of a commutative, non-associative operator
Replies: 4   Last Post: Feb 3, 2009 11:40 AM

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Dave L. Renfro

Posts: 4,792
Registered: 12/3/04
Re: A natural example of a commutative, non-associative operator
Posted: Feb 3, 2009 11:40 AM
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Mitch Harris wrote (in part):

> Is there a 'natural' example of a commutative, non-associative
> operator?

I'm not sure about 'natural', but here are some examples.

Let # denote a binary operation, whose specific definition
(including domain, which should be clear so I'll omit it)
will vary in what follows. Both [1] and [2] give the example
a#b = (a+b)/2. Item [2] goes on to give the examples
a#b = sqrt(ab), a#b = (ab)^(-1), a#b = |a - b|, and
min{a&b, b&a} where & is the operation of catenation for
positive integers written in decmial form (so 349&28 is 34,928).

Finally, [3] gives 3 examples of a binary operation on a set
each of which satisfies all the axioms of a commutative group
except associativity.

[1] Editorial Staff, "Another binary operation -- and a challenge",
Mathematics Student Journal 15 #4 (May 1968), 6.

[2] Nitsa Movshovitz-Hadar and Rina Hadass, "Between associativity
and commutativity", International Journal of Mathematical
Education in Science and Technology 12 #5 (1981), 535-539.

[3] Louis O. Kattsoff, "The independence of the associative law",
American Mathematical Monthly 65 #8 (October 1958), 620-622.

Dave L. Renfro

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