A binar is simply a set $S$ equipped with a single binary operation $\*$. A power-associative binar is a binar where the subalgebra generated by a single element is associative. Equivalently, they can be axiomatized with the infinite set of equations, $\{(x\*x)\*x=x\*(x\*x), ... \}$. Is there some finite set of axioms in the signature $\*$ that can axiomatize power-associativity?



Is the class of power-associative binars finitely axiomatizable?
No.


Indeed, let $\mathcal{V}\_n$ be the variety of magmas generated by the relating identities with variable $y$ saying that for every $k\le n$, all products of $k$ copies of $y$ are equal. Since the variety of power-associative magmas is $\bigcap\_n \mathcal{V}\_n$, a negative answer to the question is equivalent to showing that for every $n$ the relatively free magma $M\_n$ on 1-generator $x$ in $\mathcal{V}\_n$ is not associative.


Write by induction $x^1=x$, $x^k=xx^{k-1}$. I claim that in $M\_n$ we have $x^{n+1}\neq x^nx$. Indeed, $x^nx$ can be rewritten in all ways $(ab)x$ with $a,b$ products of $k,\ell$ copies of $x$ (in some order) for $k+\ell=n$. No such $(ab)x$ is a relator, and no nontrivial relator has the form $(x=\dots)$. Hence $(ab)x$ can only be transformed by a relator substitution into another $(a'b')x$. 


(This actually shows that $\mathcal{V}\_{n+1}$ is properly contained in $\mathcal{V}\_n$.)


