Quotient Structure of Monoid is Monoid

Theorem

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

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

Proof

From Quotient Structure of Semigroup is Semigroup $\struct {S / \RR, \circ_\RR}$ is a semigroup.

Let $\eqclass x {\RR} \in S / \RR$.

Consider $\eqclass e \RR$:

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

Furthermore:

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

Hence $\eqclass e \RR$ is an identity.

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

$\blacksquare$