Generated Submonoid is Intersection of Submonoids containing Generator

Theorem

Let $\struct {M, \circ}$ be a monoid whose identity is $e_M$.

Let $S \subseteq M$.

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

Then $\struct {H, \circ}$ is the intersection of all submonoids of $\struct {M, \circ}$ containing $S \cup \set {e_M}$.

Proof

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

Then by definition $H$ is the smallest (with respect to set inclusion) submonoid of $\struct {M, \circ}$ containing $S \cup \set {e_M}$.

Let $\mathbb S$ be the set of submonoids of $\struct {M, \circ}$ containing $S \cup \set {e_M}$.

We need to show that $H = \ds \bigcap \mathbb S$.

Because $H$ is a submonoid of $\struct {M, \circ}$ containing $S \cup \set {e_M}$:

$H \in \mathbb S$

By Intersection is Subset:

$\ds \bigcap \mathbb S \subseteq H$

$\Box$

By Intersection of Submonoids with Monoid Identity is Submonoid:

$\ds \bigcap \mathbb S$ is a submonoid of $\struct {M, \circ}$ containing $S \cup \set {e_M}$.

Because $H$ is the smallest (with respect to set inclusion) submonoid of $\struct {M, \circ}$ containing $S \cup \set {e_M}$:

$H \subseteq \ds \bigcap \mathbb S$

By definition of set equality:

$H = \ds \bigcap \mathbb S$

Hence the result.

$\blacksquare$

Sources

• 1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results