# Left-divisor relation in a semigroup

From Groupprops

## Contents

## Definition

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

## Facts

### The relation is transitive

This follows from associativity.

### The relation is reflexive for monoids and idempotent semigroups

If is a monoid with identity element , then is a left divisor of for all , because .

If is an idempotent semigroup (a semigroup where every element is idempotent), then is a left divisor of for every , because .

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