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.

Also see

• Definition:Generated Submonoid
• Definition:Generator of Subgroup