Intersection of Submonoids with Monoid Identity is Submonoid

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Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.

Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of submonoids of $S$.

For each $S_\alpha \in \family {S_\alpha}_{\alpha \mathop \in I}$, let $e_S \in S_\alpha$.

Let $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ denote the intersection of $\family {S_\alpha}$

Then $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is a submonoid of $S$.


First we show that $\struct {\ds \bigcap_{\alpha \mathop \in I} S_\alpha, \circ}$ is a semigroup:

Semigroup Axiom $\text S 0$: Closure

\(\ds a, b\) \(\in\) \(\ds \bigcap_{\alpha \mathop \in I} S_\alpha\)
\(\ds \leadsto \ \ \) \(\ds \forall \alpha \in I: \, \) \(\ds a, b\) \(\in\) \(\ds S_\alpha\) Definition of Intersection of Family
\(\ds \leadsto \ \ \) \(\ds \forall \alpha \in I: \, \) \(\ds a \circ b\) \(\in\) \(\ds S_\alpha\) Semigroup Axiom $\text S 0$: Closure for all $S_\alpha$
\(\ds \leadsto \ \ \) \(\ds a \circ b\) \(\in\) \(\ds \bigcap_{\alpha \mathop \in I} S_\alpha\)


Semigroup Axiom $\text S 1$: Associativity

From the above we have that $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is closed under $\circ$.

From Restriction of Associative Operation is Associative we have that $\circ$ is associative on $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$.

Hence we have that $\struct {\ds \bigcap_{\alpha \mathop \in I} S_\alpha, \circ}$ is a semigroup.


Identity Element

We are given that:

$\forall \alpha \in I: e_S \in S_\alpha$
\(\ds \forall a \in S: \, \) \(\ds a \circ e_S\) \(=\) \(\ds e_S\) Definition of Identity Element
\(\ds \forall \alpha \in I: \, \) \(\ds e_S\) \(\in\) \(\ds S_\alpha\) by hypothesis
\(\ds \leadsto \ \ \) \(\ds \forall \alpha \in I: \forall a \in S_\alpha: \, \) \(\ds a \circ e_S\) \(=\) \(\ds e_S\) by hypothesis

That is, for all $S_\alpha$, $e_S$ is the identity element of $S_\alpha$

Thus for all $S_\alpha$, $\struct {S_\alpha, \circ}$ is a monoid.