Left-divisor relation in a semigroup

Definition
Let $$(S,*)$$ be a semigroup (a set $$S$$ with associative binary operation $$*$$). The left-divisor relation in $$S$$ is defined as follows. For $$a,b \in S$$, we say that $$a$$ is a left divisor of $$b$$ if there exists $$c \in S$$ such that $$b = a * c$$.

The relation is transitive
This follows from associativity.

The relation is reflexive for monoids and idempotent semigroups
If $$S$$ is a monoid with identity element $$e$$, then $$a$$ is a left divisor of $$a$$ for all $$a$$, because $$a * e = a$$.

If $$S$$ is an idempotent semigroup (a semigroup where every element is idempotent), then $$a$$ is a left divisor of $$a$$ for every $$a \in S$$, because $$a * a = a$$.

Thus, the relation gives a quasiorder on $$S$$ when $$S$$ is a monoid or idempotent semigroup.