Quotient Structure of Monoid is Monoid

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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.

$\blacksquare$


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