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Iterating general commutative operations

  1. Feb 12, 2009 #1
    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?
  2. jcsd
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