# 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.