Quotient Structure of Semigroup is Semigroup

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Theorem

Let $\mathcal R$ be a congruence relation on a semigroup $\left({S, \circ}\right)$.


Then the quotient structure $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a semigroup.


Proof

From Quotient Structure is Well-Defined we have that $\circ_\mathcal R$ is closed on $S / \mathcal R$.

Let $\left[\!\left[{x}\right]\!\right]_\mathcal R, \left[\!\left[{y}\right]\!\right]_\mathcal R, \left[\!\left[{z}\right]\!\right]_\mathcal R \in S / \mathcal R$.

We shall prove that $\circ_\mathcal R$ is associative:

\(\displaystyle \left({\left[\!\left[{x}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{y}\right]\!\right]_\mathcal R}\right) \circ_{S / \mathcal R} \left[\!\left[{z}\right]\!\right]_\mathcal R\) \(=\) \(\displaystyle \left[\!\left[{x \circ y}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{z}\right]\!\right]_\mathcal R\) Definition of operation induced on $S / \mathcal R$ by $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{ \left({x \circ y}\right) \circ z }\right]\!\right]_\mathcal R\) Definition of operation induced on $S / \mathcal R$ by $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{ x \circ \left({y \circ z}\right) }\right]\!\right]_\mathcal R\) $\circ$ is Associative
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{x}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{y \circ z}\right]\!\right]_\mathcal R\) Definition of operation induced on $S / \mathcal R$ by $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{x}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left({\left[\!\left[{y}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{z}\right]\!\right]_\mathcal R}\right)\) Definition of operation induced on $S / \mathcal R$ by $\circ$

Hence $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a semigroup.


$\blacksquare$