\begin{definition}[Definition:Generated Submonoid]
Let $\struct {M, \circ}$ be a monoid whose identity is $e_M$.
Let $S \subseteq M$
Let $H$ be the smallest (with respect to set inclusion) submonoid of $M$ such that $\paren {S \cup \set {e_M} } \subseteq H$.
Then $\struct {H, \circ}$ 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}