# Intersection of Submonoids with Monoid Identity is Submonoid

## Theorem

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 $S$.

## Proof

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 \ \$ $\ds a \circ b$ $\in$ $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$

$\Box$

### 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.

$\Box$

### 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.

$\blacksquare$