Is this an axiomatization of the meet of the commutative and associative theories?

Question

This is related to a question I asked before. In that question, I asked if the meet of the commutative and associative properties can be axiomatized by the equation $(x * y) * x = x * (y * x)$. The answer was no. So, now I am asking, if we add also the equation, $((x*y)*z)*(x*(y*z))=(x*(y*z))*((x*y)*z)$ to the previous equation, would we get an axiomatization of the meet. It would be preferable if someone could give a finite axiomatization, or prove that there is none, but that is optional for this particular question.

Answer 1

Consider the algebras with the following multiplication tables:

The first is commutative, but not associative, since $$(0 \cdot 0) \cdot 1 = 1 \cdot 1 = 0 \neq 1 = 0 \cdot 0 = 0 \cdot (0 \cdot 1).$$ On the other hand, being commutative, it trivially satisfies your equalities.

The second is the two-element left-zero semigroup, defined by $x\cdot y = x$.
So it trivially satisfies your equalities (since the left most variable is the same on both sides of bot equalities).
Clearly, it is not commutative.

Hence, those two equalities combined don't entail neither associativity nor commutativity.