# Is the class of power-associative binars finitely axiomatizable?

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?

• I can think of some two variable identities that would imply power associativity, but would be strictly stronger. Are you sure a finite basis exists? Gerhard "Trying Not To Get Hyper" Paseman, 2020.06.06. – Gerhard Paseman Jun 6 at 21:55
• Try the following for a counterexample. Let an algebra be generated by x and have standard power arithmetic up to n. Then, whenever two items have exponents which sum to a prime bigger than n, let the kth power times the jth power be equal to a symbolic term x to the power (k,j). In multiplying these, you can collapse the products as you like, as long as you keep (k,j) distinct from (j,k). Gerhard "Always Look For Prime Examples" Paseman, 2020.06.06. – Gerhard Paseman Jun 6 at 22:40

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 $$\mathcal{V}_n$$.)

• In case it helps, here is a pedestrian definition of $M_k$. Let $J_1(x)$ be the free magma on 1 generator $x$ with chosen total order; enumerate pairs $(P_i,Q_i)$ of distinct elements of $J_1$ with $P_i<Q_i$ and $P_i,Q_i$ of the same size; choose the enumeration in increasing order of size, e.g. $P_0=x(xx)$, $Q_0=(xx)x$, $P_1=x(x(xx))$, $Q_1=(xx)(xx)$, $P_2=(xx)(xx)$, $Q_2=((xx)x)x$, etc. Write $i_n=\max\{i:|P_i|\le n\}$.Then $M_n$ is the quotient of $J_1$ by the magma equivalence relation (= quotient inherits magma stricture) generated by $P_i(z)=Q_i(z)$ for all $z\in J_1$. – YCor Jun 7 at 10:44
• In the comment, you should add "...for $i\leqslant i_n$", right? – მამუკა ჯიბლაძე Jun 7 at 19:17
• @მამუკაჯიბლაძე Yes, sorry, of course this why I introduced $i_n$. – YCor Jun 7 at 19:19
• @მამუკაჯიბლაძე no it's correct (and correcting as you're suggesting would be correct too): for a 1-generated magma, associative and power-associative are the same by definition. I understand user158834's objection as understanding that $M_n$ is defined as the magma with finite presentation: 1 generator $x$, and, as relators, all associative relations of length $\le n$ w.r.t. $x$ (instead of taking also substitutions using these relations). – YCor Jun 7 at 20:22
• PS: the answer by user158834 was downvoted because of its initial tone, but what remains of it now is very interesting information. – YCor Jun 17 at 14:19

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.

• I recall remarking to George Bergman around that time that there was a previous usage, from Star Trek TNG. I think he had the idea before the conference. Gerhard "Decided Not To Use Klingon" Paseman, 2020.06.07. – Gerhard Paseman Jun 8 at 2:47
• I've asked J-P. Serre about the invention of the term magma: his answer is "Il me semble que c'est Bourbaki qui a inventé ça. Autant que je souvienne, il voulait un mot qui n'ait pas été déjà utilisé en algèbre, et qui suggère quelque chose n'ayant aucune structure intéressante. Je crois que nous avions pensé à "fourbi", mais nous l'avons rejeté. En tout cas ça n'a rien à voir avec Ore." English rough translation: "I think Bourbaki invented this. As far as I know, he wanted a word not yet used in algebra, suggesting something that has no interesting structure. (...) – YCor Jun 9 at 7:09
• (...) I think we thought of "fourbi" but we rejected it. In any case it's unrelated to Ore" – YCor Jun 9 at 7:10
• @YCor Thanks for thi comment. Since I learned the word "magma" I intuitively thought of it as a kind of primordial chaos. I was appalled by the explanation in this answer, and I'm glad to know that it's not true. – bof Jun 10 at 4:37
• The m-word sounds like an awful mockery of a name whether or not it is related to Ore. Also, “something that has no interesting structure” (primordial chaos) completely misses the point: Bourbaki, in their infinite wisdom, were apparently unaware that algebraists are not so much interested in the class of all groupoids (there is not much to say about it in this generality, and what is there to say also applies to algebras in other signatures), but they usual study varieties of groupoids satisfying this or that additional identity, precisely because they do have interesting structure. – Emil Jeřábek Jun 10 at 6:46

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.

• Could you elaborate? At present this answer isn't very useful. – Noah Schweber Jun 6 at 22:59
• "My answer appeared 30 min earlier than the other one" Yes, but until 22 minutes ago - significantly after the other answer - your answer merely consisted of the first two sentences. So that's rather disingenuous. – Noah Schweber Jun 7 at 0:03
• You're claiming that my answer is false because I'm not considering the right magma. No, I mean the free object in $V_n$ (that's why I wrote $y$ for the letter in the identity and $x$ the generator of the magma). So in $M_n$ I really mean all relators, e.g. in $V_3$ relators include not only $(xx)x=x(xx)$, but also $((xx)(xx))(xx)=(xx)((xx)(xx))$, etc, i.e., obtained from relating identities by replacing $y$ by an arbitrary word in $x$. I wrote "varieties generated by relators..." because I think of it as relators and it generates with the rules of varieties, including substitution. – YCor Jun 7 at 7:33
• The two different usages of groupoids is well-entrenched, and unlikely to go away. Brandt's original definition is much closer to the category-theoretic definition, so it's definitely not true that they hijacked the universal-algebraic definition, but the UA definition is almost as old. If UAists want to spontaneously change to "magma" or "binar", then that's fine, but you sometimes see an attempt to impose a name change from the outside, which is unlikely to work. – arsmath Jun 7 at 8:23
• @მამუკაჯიბლაძე I'm sorry I'm not familiar with "commonly used" terminology (if any); in any case I can't imagine any other sensible meaning to "variety generated by the relating identities (...)" other than the relevant one. Namely, given a set $X$ of "relating identities" (a set of equalities between elements of free magmas, e.g. $\{x(yz)\equiv (xy)z,xy\equiv yx\}$), by "the variety generated (defined?) by the relating identities $X$" I mean the collection of all magmas in which all these identities hold. – YCor Jun 7 at 10:30