Finite algebraic structure where there is no finite generating set of equations

Question

Let $A$ be an algebra whose carrier set is finite. Must it be the case that there is a finite set of equations which generate all the universally valid equations in that structure? If not, can anyone give me a counterexample?

Answer 1

No. Let $M = \{1, a, b, ab, ba, 0\}$ be the monoid defined by the relations $aba = a$, $bab = b$, $aa = bb = 0$ and of course $1x = x1 = x$ and $0x = x0 = 0$ for all $x \in M$. It was proved by Perkins that the variety of monoids generated by $M$ is not finitely based.

[1] P. Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298–314.