# Quotient Structure of Monoid is Monoid

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## 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.

$\blacksquare$

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Theorem $1.5$