# Quotient Structure of Monoid is Monoid

## Theorem

Let $\mathcal R$ be a congruence relation on a monoid $\struct {S, \circ}$ with an identity $e$.

Then the quotient structure $\struct {S / \mathcal R, \circ_\mathcal R}$ is a monoid.

## Proof

From Quotient Structure of Semigroup is Semigroup $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a semigroup.

Let $\eqclass x {\mathcal R} \in S / \mathcal R$.

Consider $\eqclass e {\mathcal R}$:

 $\displaystyle \eqclass x {\mathcal R} \circ_{S / \mathcal R} \eqclass e {\mathcal R}$ $=$ $\displaystyle \eqclass {x \circ e} {\mathcal R}$ Definition of Operation Induced on $S / \mathcal R$ by $\circ$ $\displaystyle$ $=$ $\displaystyle \eqclass x {\mathcal R}$ Definition of Identity Element

Furthermore:

 $\displaystyle \eqclass e {\mathcal R} \circ_{S / \mathcal R} \eqclass x {\mathcal R}$ $=$ $\displaystyle \eqclass {e \circ x} {\mathcal R}$ Definition of Operation Induced on $S / \mathcal R$ by $\circ$ $\displaystyle$ $=$ $\displaystyle \left[\!\left[{x}\right]\!\right]_\mathcal R$ Definition of Identity Element

Hence $\eqclass e {\mathcal R}$ is an identity.

Hence $\struct {S / \mathcal R, \circ_\mathcal R}$ is a monoid.

$\blacksquare$