# Definition:Generated Submonoid

## Definition

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 $H = \gen {\set x}$.

### Generator

Let $\struct {H, \circ}$ be the submonoid of $\struct {M, \circ}$ generated by $S$.

Then $S$ is known as a generator of $\struct {H, \circ}$.