Definition:Generator of Subsemigroup

Definition
Let $\struct {S, \circ}$ be a semigroup.

Let $\O \subset X \subseteq S$.

Let $\struct {T, \circ}$ be the smallest subsemigroup of $\struct {S, \circ}$ such that $X \subseteq T$.

Then:
 * $X$ is a generator of $\struct {T, \circ}$
 * $X$ generates $\struct {T, \circ}$
 * $\struct {T, \circ}$ is the subsemigroup of $\struct {S, \circ}$ generated by $X$.

This is written:
 * $T = \gen X$

Also known as
Some sources refer to such an $X$ as a set of generators of $T$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $X$ is itself a generator of $T$ independently of the other elements.

Also see

 * Existence of Unique Subsemigroup Generated by Subset