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 $S \cup \set {e_M}$:
- $H \in \mathbb S$
- $\ds \bigcap \mathbb S \subseteq H$
$\Box$
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