# Quotient Structure of Semigroup is Semigroup

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## Theorem

Let $\RR$ be a congruence relation on a semigroup $\struct {S, \circ}$.

Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a semigroup.

## Proof

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

Let $\eqclass x \RR, \eqclass y \RR, \eqclass z \RR \in S / \RR$.

We shall prove that $\circ_\RR$ is associative:

\(\displaystyle \paren {\eqclass x \RR \circ_{S / \RR} \eqclass y \RR} \circ_{S / \RR} \eqclass z \RR\) | \(=\) | \(\displaystyle \eqclass {x \circ y} \RR \circ_{S / \RR} \eqclass z \RR\) | Definition of Operation Induced on $S / \RR$ by $\circ$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {\paren {x \circ y} \circ z} \RR\) | Definition of Operation Induced on $S / \RR$ by $\circ$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {x \circ \paren {y \circ z} } \RR\) | $\circ$ is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass x \RR \circ_{S / \RR} \eqclass {y \circ z} \RR\) | Definition of Operation Induced on $S / \RR$ by $\circ$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass x \RR \circ_{S / \RR} \paren {\eqclass y \RR \circ_{S / \RR} \eqclass z \RR}\) | Definition of Operation Induced on $S / \RR$ by $\circ$ |

Hence $\struct {S / \RR, \circ_\RR}$ is a semigroup.

$\blacksquare$