A finite axiomatization of the join of the commutative and associative properties

Question

Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by commutativity and the theory axiomatized by associativity is simply the theory axiomatized by both of them. What about the join? Is there a finite axiomatization of the join? I conjecture that it is axiomatized by the equation, $(x*y)*x=x*(y*x)$. Is this true? If not, is there some other finite axiomatization, and if so, can someone exhibit such a finite axiom set.

Answer 1

Partial answer: I don't know about a finite equational basis, but the "flexible law" $$(x*y)*x=x*(y*x)\tag1$$ is insufficient: the identity $$((x*y)*z)*(x*(y*z))=(x*(y*z))*((x*y)*z)\tag2$$ follows from the associative law, and it follows from the commutative law, but it does not follow from $(1)$.

Let $\mathbb F$ be any field not of characteristic $2$ or $3$. If we define $$x*y=\frac23x+\frac13y$$ then the magma $(\mathbb F,*)$ satisfies $(1)$ for all $x$ and $y$, but does not satisfy $(2)$ if $z\ne x$.

Note that $$x*y=y*x\iff x=y.$$ Now $$(x*y)*z=\frac23\left(\frac23x+\frac13y\right)+\frac13z=\frac49x+\frac29y+\frac39z$$ and $$x*(y*z)=\frac23x+\frac13\left(\frac23y+\frac13z\right)=\frac69x+\frac29y+\frac19z,$$ so $$(x*y)*z=x*(y*z)\iff z=x,$$ so $$(x*y)*x=x*(y*x),$$ that is, $(1)$ holds. Also $$((x*y)*z)*(x*(y*z))=(x*(y*z))*((x*y)*z)$$$$\iff(x*y)*z=x*(y*z)\iff z=x,$$ so $(2)$ does not hold unless $z=x$.