A natural example of a commutative, non-associative operator

[Mitch Harris]

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

The canonical example of an associative, non-commutative operator is
matrix multiplication, but for the life of me I can't think of the
other kind.

My imagination tells me something to do with unordered binary trees,
but I can't think of a domain where that naturally fits.

Mitch

[tc...@lsa.umich.edu]

In article <108ee92f-a8e0-4eeb...@p36g2000prp.googlegroups.com>,

Mitch Harris <mah...@gmail.com> wrote:
>Is there a 'natural' example of a commutative, non-associative operator?

Look up "Jordan algebras."
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

[Hagen]

Genetic inheritance can be described algebraically
using so-called genetic algebras, which are commutative
but not associative.
See for example:

www.ams.org/bull/1997-34-02/S0273-0979-97-00712-X/S0273-0979-97-00712-X.pdf

H

[Jack Schmidt]

>> Is there a 'natural' example of a commutative,
>> non-associative operator?
>
> Look up "Jordan algebras."

A special case of Jordan algebras are Lie algebras in
characteristic 2. They might be more familiar. In
general, for an associative algebra, the operation
that takes two elements x and y and returns xy+yx
is a Jordan algebra, but there are other exceptional
Jordan algebras as well.

[Dave L. Renfro]

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