Left-divisor relation in a semigroup
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 .
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.