TITLE: Is the class of power-associative binars finitely axiomatizable?
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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:

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.