An example of a binary operation which is neither idempotent nor has a right identity, but has a reflexive "left-divisor" relation.

Question

Let $A$ be a set with a binary operation $*$. I define a binary relation $R$ on $A$ by defining $xRy$ to hold if there exists a $z$ in $A$ such that $x*z=y$, and in that case I call $x$ a "left-divisor" of $y$. If $(A,*)$ has a right identity, then the relation $R$ is reflexive. Also, if $(A,*)$ is an idempotent binary operation, then the relation $R$ is reflexive. I want an example of a finite non-empty magma $(A,*)$ such that the associated relation $R$ is reflexive, but such that $(A,*)$ is neither idempotent nor has a right identity, if such a magma exists.