\begin{definition}[Definition:Generator of Subsemigroup]
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:
:$T = \gen X$
\end{definition}