What is the property where f(g(x)) = g(f(x))?
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J. M. ain't a mathematician
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asked Jan 12, 2011 at 3:52
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2$\begingroup$ Here is a related post I made a while back: math.stackexchange.com/questions/11431/… $\endgroup$– AnonymousCowardCommented Jan 12, 2011 at 6:13
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$\begingroup$ Hmm, I seem to have slipped up with the rollbacks here; on the other hand, it doesn't seem right that the "associative" part in the previous iteration of the question is missing... $\endgroup$– J. M. ain't a mathematicianCommented Apr 29, 2011 at 18:56
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2 Answers
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Besides being called (composition) commutative, it is sometimes also said that such functions are permutable, e.g. see here. As an example, a classic result of Ritt shows that permutable polynomials are, up to a linear homeomorphism, either both powers of x, both iterates of the same polynomial, or both Chebychev polynomials.
answered Jan 12, 2011 at 4:57
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We say $f$ and $g$ commute (with respect to composition). The property is called "commutativity".
"Associativity" is the property that says that $f\circ (g\circ h)$ is the same as $(f\circ g)\circ h$, where $\circ$ is composition.
answered Jan 12, 2011 at 3:54