Is the class of power-associative binars finitely axiomatizable?

Question

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?

Answer 1

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 $\mathcal{V}_n$.)

Answer 2

The question has been answered, but I will add some remarks about magma/groupoid/binar. This is in response to some of the comments on this page:

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What you can currently read on the English Wikipedia:

The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid (translated from the German Gruppoid). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. … The term magma was used by Serre [Lie Algebras and Lie Groups, 1965]. It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.

What you can currently read on the French Wikipedia:

L'ancienne appellation « groupoïde de Ore », introduite par Bernard Hausmann et Øystein Ore en 1937 et parfois utilisée jusque dans les années 1960, est aujourd'hui à éviter, l'usage du terme groupoïde étant aujourd'hui réservé à la théorie des catégories, où il signifie autre chose. [The old name groupoïde de Ore, introduced by Bernard Hausmann and Øystein Ore in 1937 and sometimes used until the 1960s, is to be avoided today, the use of the term groupoid being today reserved for category theory, where it means something else.]

What is currently not on either Wikipedia:

A. (magma) Peter Shor of MIT has speculated that magma might have been introduced as a pun on Ore's name. To distinguish a groupoid in the sense of Ore from a groupoid in the sense of Brandt, the phrase groupoid of Ore was used first, then shortened to magma, which in geology means a pile of molten ore.

B. (binar) In June 1993, there was a conference at MSRI on Universal Algebra and Category Theory organized by Ralph McKenzie and Saunders Mac Lane. At this conference, there was a discussion session where several topics were discussed, including whether there could be general agreement on the future use of the word groupoid. At this discussion session, George Bergman of Berkeley proposed using binar to mean an algebraic structure with a single binary operation. (He also proposed unar for an algebraic structure with a single unary operation.) It seemed to me that Bergman thought this up on the spot, so it is plausible that he is the source of binar.

Answer 3

No this class is not finitely axiomatzable. It can be easily proved by standard methods.

Update 1. For non-specialists. One can use the much stronger result from Gaĭnov, A. T. Power associative algebras over a field of finite characteristic.
Algebra i Logika 9 1970 9–33. Gainov proved that if $F$ is a field of prime characteristic (say, $F_p$) then the variety of all power associative $F$-algebras is not finitely axiomatizable (note that for characterstic 0 Albert's result from 1948 shows that the variety is finitely based). This means that for every $n$ there exists an associativity identity $u=v$ in one variable and an $F$-algebra $S$ which satisfies all asdociativity identities of length $\le n$ but does not satisfy $u=v$. The multiplicative magma of $S$ has the same property because the identities do not involve additions. Thus the variety of power associative magmas (which the OP calls binars for some unknown reasons) is not finitely based.