Does there exist a commutative magma such that $\mid$ is transitive, which is not a semigroup?

Question

Let $M$ denote a commutative magma, and write $x \mid y$ iff $xa=y$ for some $a \in M$. If $M$ is a semigroup, then $\mid$ is transitive. Does there exist a commutative magma such that $\mid$ is transitive, but which is not a semigroup?

I'm guessing "yes." (But can't think of any examples).

Answer 1

$M=\{a,b,c\},aa=bc=cb=a,bb=ac=ca=b,cc=ab=ba=c$.

Moreover, any commutative magma can be embedded in a commutative magma in which $\mid$ is transitive.