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Re: A natural example of a commutative, nonassociative operator
Posted:
Feb 3, 2009 11:40 AM


Mitch Harris wrote (in part):
> Is there a 'natural' example of a commutative, nonassociative > 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 MovshovitzHadar and Rina Hadass, "Between associativity and commutativity", International Journal of Mathematical Education in Science and Technology 12 #5 (1981), 535539.
[3] Louis O. Kattsoff, "The independence of the associative law", American Mathematical Monthly 65 #8 (October 1958), 620622.
Dave L. Renfro



