This question already has an answer here:
The hyperoperation heirarchy in the naturals starts with addition, then multiplication, then exponentiation, then tetration, and so on. Each operation is defined as repeated application of the previous in the sequence. Here, we are defining 0^0, 0^^0, etc, x^0, x^^0, etc as 1. In a discussion on some online forum, someone was remarking that exponentiation satisfies the identity (x^y)^z=(x^z)^y but tetration does not. And of course, exponentiation fails to satisfy the commutative law. He wondered if this process keeps on happening as you ascend the hyperoperation heirarchy. So, what I would like to know is if the sequence of sets of identities of the hyperoperation heirarchy starting from multiplication, is strictly descending. By identities I mean universally quantified equations without constants. If anyone can link me to a paper on this or a similar topic, I would be immensely delighted.