Power-associative magma

In terms of the existence of powers
Suppose $$(S,*)$$ is a magma, i.e., $$S$$ is a set and $$*$$ is a binary operation on $$S$$. We say that $$(S,*)$$ is power-associative (or equivalently, that $$*$$ is power-associative on $$S$$) if we can define a power function:

$$S \times \mathbb{N} \to S$$

denoted by $$(x,n) \mapsto x^n$$, such that:

$$x^1 = x \ \forall x \in S$$

and

$$\! x^m * x^n = x^{m + n} \ \forall x \in S, m,n \in \mathbb{N}$$.

If such a function exists, it is unique and $$x^n$$ equals the value of any $$n$$-fold product of $$x$$, no matter how it is parenthesized.

Other definitions
A power-associative magma is a magma satisfying the following equivalent conditions:


 * 1) It is expressible as the union of subsemigroups, i.e., submagmas that are associative under the operation.
 * 2) It is expressible as the union of abelian subsemigroups, i.e., submagmas that are associative and commutative under the operation.