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 $b=a*c$.

Facts

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.