Is there a finite equational basis for the join of the commutative and associative equations?

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?