Two questions regarding equational axiomatizations of power-associative magmas.

Question

A power-associative magma is a magma $(M;*)$ where the submagma generated by any single $x$ in $M$, is associative. I have two questions regarding power-associative magmas. First, some terminology. Let $t$ be a term, in the sense of universal algebra, and let $m$ be a positive integer. I recursively define $t^m$ by $t^1=t$ and $t^{m+1}=t^m * t$. My first question is, does the collection of equations $x^m * x^n = x^{m+n}$, where $m,n$ range over all positive integers, axiomatize the class of power-associative magmas? And my second question is, does the collection of equations $(x^m)^n = x^{mn}$, where $m,n$ range over all positive integers, axiomatize the class of power-associative magmas? If the answer to either question is no, can someone exhibit a magma which satisfies those weaker equational theories, but is not power-associative?

Answer 1

The answer is yes for $x^m*x^n=x^{m+n}$, no for $(x^m)^n=x^{mn}$.

For fixed $x$ the identities $x^m*x^n=x^{m+n}$ say that the map $n\mapsto x^n$ is a homomorphism from the additive semigroup of positive integers to the submagma generated by $x$, which is therefore associative.

Define $x*y$ for $x,y\in\mathbb N$ so that $x*y=x+y\iff x\ge y$; then the magma $(\mathbb N,*)$ satisfies the identities $(x^m)^n=x^{mn}$, while $x*(x*x)\ne(x*x)*x$.