# Definition:Generator of Monoid

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## Definition

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

## Also known as

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