Let $(A, +, 0_A, \cdot, 1_A)$ be a commutative ring with a unity. And, let $(S, \cdot, 1_S)$ be a commutative monoid.
An "exponentiation" is an operation $S\times A\to S,$ where $(x, m)\mapsto x^m,$ possibly partial in second variable satisfying the properties:
$$x^{0_A}=1_S,\quad x^{1_A}=x, \quad x^{m+n}=x^mx^n,\quad (x^m)^n=x^{mn},\quad (xy)^m=x^my^m.$$
Suppose we reduced some structure and choose $A=\mathbb{N}.$ Then the minimal structure we need to define non-negative integer exponents is just a commutative monoid $S.$
In order to have negative integer exponents, in which case $A=\mathbb{Z},$ we need (multiplicative) inverses. So, we need $S$ to be at least an abelian group.
Here I am wondering, what if we need the exponents also in side the algebraic structure $S$? That is $A=S.$
Also, if we need "exponentiation" coincides with "integer exponents", then $S$ must be a ring of characteristic zero. But I am not sure how to define $a^b$ with $a,b\in S$ in this generality. Any thoughts?