# How natural is the associativity law, or are there any real world non-associative examples

Are there any interpretations and applications of algebraic structures which are not associative?

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en.wikipedia.org/wiki/Octonion – xavierm02 Nov 30 '13 at 12:42
nice, I am also looking for ''real world'' examples which could be given to a layman, like explaining commutativity by saying taking on clothes is non-commutative, but ordering a deck of cards is commutative because it does not matters in what order you perform your tasks – Stefan Nov 30 '13 at 12:46

What about subtraction? We have $x-(y-z) \neq (x-y)-z$ when $z \neq 0$.

Example from real world ;-): You have 5 apples, and would like to give 3 of them to 3 friends. But 1 of them doesn't like it. So in the end you have $5-(3-1)=3$ apples, and not $(5-3)-1=1$.

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And division, of course. $16/(4/2)\neq (16/4)/2$. – Christopher Creutzig Nov 30 '13 at 12:49
lmao the real world example is just too much! – user88576 Nov 30 '13 at 12:49
roflcopter your nickname (SCNR) – Martin Brandenburg Nov 30 '13 at 12:51
Three diners have to pay a bill for a meal. They pay ten dollars each, making $30$ dollars, but the cost was really only twenty-five dollars in total, so the waiter pockets two dollars and returns one to each diner. So the diners end up paying a net nine dollars each, making $27$, and the waiter now has $2$, making $29$, and the remaining dollar of the original $30$ has disappeared into non-associativity. – Henry Nov 30 '13 at 16:33

The arithmetic mean $a*b=\frac{a+b}{2}$ is a medial operation: $(a*b)*(c*d)=(a*c)*(b*d)$. This property is used in psychophysics.

See J. Pfanzagl, Theory of Measurement, 2nd ed., Physica-Verlag, Wurzburg—Wien, 1971, chapt.7.2 (Pfanzagl named such an operation bisymmetric).

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do you remember what specifically is modelled with it in psychophysics, as I know psychophysics deals with the mapping of stimuli (real physical entities) to sensations (perceived ''quantity'' of stimulus), like air pressure to loudness, light waves to color and brightness, masses to ''heaviness'' and so on. – Stefan Nov 30 '13 at 15:10
I wrote a reference in the question. – Boris Novikov Nov 30 '13 at 15:44
Not just the arithmetic mean. An $f$-mean has this property – Henry Nov 30 '13 at 16:26
@Henry Of course. And the geometric mean also. – Boris Novikov Nov 30 '13 at 16:29
@Boris - the geometric mean is an $f$-mean with $f(x)=\log(x)$ – Henry Nov 30 '13 at 16:35