I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the commutative equation and theory axiomatized by the associative equation is the theory axiomatized by both of them. What about the join? Is there a finite equational basis for the join of those theories?

$\begingroup$ For this example, I don't know. I recall some work about chains of varieties, probably semi group varieties ,where every other member was not finitely based. I suspect nfb (and thus fb) is not well behaved under join. I don't recall the author names, but I would be unsurprised if one of them was Mark Sapir. Gerhard "Look Up Chains And NFB" Paseman, 2020.07.14. $\endgroup$ – Gerhard Paseman Jul 14 at 18:19

3$\begingroup$ If I understand this right, one theory in the join is the theory of $a(ba)=(ab)a$, since that follows from either commutativity or associativity. Are there other interesting theories in this join? $\endgroup$ – Matt F. Jul 14 at 23:21

3$\begingroup$ @MattF. $((ab)c)(a(bc))=(a(bc))((ab)c)$ at least if you don't insist on "interesting". $\endgroup$ – bof Jul 15 at 0:25

4$\begingroup$ @bof, thanks; that’s good enough to make me think the answer to the question is no. $\endgroup$ – Matt F. Jul 15 at 1:05

5$\begingroup$ Here's a purely algebraic (rough) formulation of the question: roughly: what are identities satisfied by all associative magmas, and by all commutative magmas? is there a finite number of identities generating all those? $\endgroup$ – YCor Jul 15 at 7:32