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 $\eqclass e \RR$:

\(\ds \eqclass x \RR \circ_{S / \RR} \eqclass e \RR\)

\(=\)

\(\ds \eqclass {x \circ e} \RR\)

Definition of Operation Induced on $S / \RR$ by $\circ$

\(\ds \)

\(=\)

\(\ds \eqclass x \RR\)

Definition of Identity Element

Furthermore:

\(\ds \eqclass e \RR \circ_{S / \RR} \eqclass x \RR\)

\(=\)

\(\ds \eqclass {e \circ x} \RR\)

Definition of Operation Induced on $S / \RR$ by $\circ$

\(\ds \)

\(=\)

\(\ds \eqclass x \RR\)

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 $1.5$