What nice properties does exponentiation have?

Exponentiation of course satisfies a number of nontrivial identities:

• $x^{y+z}=x^yx^z$

• $(x^y)^z=x^{yz}$

• $x^0=1$, $x^1=x$

However, these identities all involve functions other than exponentiation (I'm thinking of $0$ and $1$ as nullary functions, here). My question is what identities hold of exponentiation alone. That is:

What is the equational theory of $(\mathbb{N}, exp)$?

To be clear, I mean "identity" in the strict, universal-algebraic sense: one term equals another term, where each term is built from variables and exponentiation alone. Also, an identity has to hold on all of $\mathbb{N}$: identities which hold only on, say, numbers divisible by $17$ don't count.

A related question:

Is that theory axiomatized by finitely many equations?

Note: A previous version of this question asked whether there were any nontrivial identities at all. This was extremely silly of me, as pointed out almost immediately by Stefan Perko below: $(x^y)^z=(x^z)^y$.

Solution 1:

I learned indirectly that Martin [1] showed that the identity $(x^z)^y = (x^y)^z$ is complete for the standard model ⟨N, ↑⟩ of positive natural numbers with exponentiation. Unfortunately, I don't have access to this article. Could someone confirm this information?

[1] Charles F. Martin. Axiomatic bases for equational theories of natural numbers. Notices of the Am. Math. Soc., 19(7), 778 (1972).