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.

$\blacksquare$


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