Two finitely based equational theories whose meet is not finitely based.

Question

Consider the lattice of equational theories of a single binary operation $*$. The join of two finitely based equational theories is of course finitely based. Do there exist two finitely based equational theories whose meet is not finitely based?

Answer 1

Such theories exist. I will describe a construction from the paper

Finitely Based, Finite Sets of Words.
M. Jackson, O. Sapir,
International Journal of Algebra and Computation 10(6):683-708 (2000).

Let $X=\{a,b\}$ and let $M(a,b)$ be the free monoid over $X$. If $W\subseteq X^*$ is a set of words in the letters $X$, let $I(W)$ be the ideal of $M(a,b)$ consisting of all non-identity monoid elements that are not subwords of words in $W$. Define $S(W) = M(a,b)/I(W)$.

In Theorem 5.8 of their paper, Jackson and Sapir show that

(i) $A:=S(\{abbaa, ababa, aabba\})$ is finitely based.
(ii) $B:=S(\{baaab, aabb, abba, abab\})$ is finitely based.
(iii) $A\times B$ is not finitely based.

This answers the question because $\textrm{Th}(A\times B)=\textrm{Th}(A)\cap \textrm{Th}(B)$.