Left-divisor relation in a semigroup

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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.