\begin{definition}[Definition:Generator of Monoid]
Let $\struct {M, \circ}$ be a monoid.
Let $S \subseteq M$.
Let $H$ be the smallest submonoid of $M$ such that $S \subseteq H$.
Then:
:$S$ is a '''generator''' of $\struct {H, \circ}$
:$S$ '''generates''' $\struct {H, \circ}$
:$\struct {H, \circ}$ is the '''submonoid of $\struct {M, \circ}$ generated by $S$'''.
This is written $H = \gen S$.
If $S$ is a singleton, for example $S = \set x$, then we can (and usually do) write $H = \gen x$ for $H = \gen {\set x}$.
\end{definition}