# Iterating general commutative operations

1. Feb 12, 2009

### Preno

So multiplication is an iteration of addition and exponentiation is the iteration of multiplication, i.e. a "second" iteration of multiplication (let's say it's a "right-iteration", i.e. (a+(a+a))). Addition is commutative, its first iteration is commutative, but its second iteration isn't. My question is: given an arbitrary commutative operation @, if we continue iterating it, do we always eventually get a non-commutative operation? If no, is there a fixed n such that for some i<n, the i-th iteration of every commutative operation is non-commutative?

(For a*b+2 and a^b + b^a, the first iteration is already non-commutative.)

Also, is it possible to obtain a commutative operation by iterating a non-commutative one?