The ratio of finitely based magmas to all magmas

Question

Let $n$ be a positive integer. By $S_n$, I denote the set of positive integers from $1$ to $n$. By $F_n$, I denote the cardinality of the set of magmas on $S_n$ which are finitely based, that is, which have a finite generating set of identities.

I conjecture that the ratio of $F_n$ to the cardinality of the set of all magmas on $S_n$ tends to $0$ as $n$ tends to infinity.

Is this true, and if not, what is the ratio? And has anyone wrote a paper on this topic?

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Non-OP edit:

Here's the precise definition of "finitely based" (since there's some potential confusion around what "generating set of identities" means - under one interpretation it trivially includes all finite algebras):

An algebra $A$ is finitely based iff its equational theory can be axiomatized by finitely many equations (where "equation" is meant in the sense of universal algebra). Equivalently, iff there is a finite set of equations $F$ such that the variety $Mod(F)$ of algebras satisfying each equation in $F$ is exactly the variety generated by $A$. There do indeed exist non-finitely-based finite algebras (including a three-element magma), and in fact the general problem of determining whether a finite algebra is finitely based is extremely complicated - see e.g. here.

Answer 1

A magma $\mathbf{A}$ is primal if for every $n\geq 1$, every operation $A^n\to A$ can arise from a word on $n$ letters. Let $P_n$ be the cardinality of the set of all primal magmas on $S_n$. Let $M_n$ be the cardinality of the set of all magmas on $S_n$.

Theorem $$\lim_{n\to\infty}\frac{P_n}{M_n}=1/e\approx 0.368$$

This was supposedly originally proved by R.O. Davies in 1968 and strengthened by V.L. Murskii in the 1970s. I am only familiar with the treatment in Cliff Bergman's "Universal Algebra: Fundamentals and Selected Topics" however.

If every operation can arise as a term operation, then necessarily a primal magma has a sequence of Jonsson operations, ensuring that it generates a congruence distibutive variety (see "Algebras whose congruence lattices are distributive". Bjarni Jonsson. Mathematica Scandinavica, 1968.)

If a finite magma generates a congruence distributive variety, then it must be finitely based (see "Finite equational bases for finite algebras in a congruence-distributive equational class". Kirby Baker. Advances in Mathematics, 1977.)

Hence $$\lim_{n\to\infty}\frac{F_n}{M_n}\geq\lim_{n\to\infty}\frac{P_n}{M_n}=1/e>0.$$

Additional Comment (Thanks to amsra)

An operation $A^n\to A$ is idempotent if $(a,a,\dots,a)\mapsto a$ for all $a\in A$. A magma $\mathbf{A}$ is idemprimal if for every $n\geq 1$, every idempotent operation $A^n\to A$ can arise from a word on $n$ letters. Let $I_n$ be the cardinality of the set of all idemprimal magmas on $S_n$. Then Murskii's result was that:

$$\lim_{n\to\infty}\frac{I_n}{M_n}=1.$$

Since Jonsson operations need to be idempotent, it follows from the above argument that

$$\lim_{n\to\infty}\frac{F_n}{M_n}=1.$$