Quotient Structure of Semigroup is Semigroup

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:

\(\ds \paren {\eqclass x \RR \circ_{S / \RR} \eqclass y \RR} \circ_{S / \RR} \eqclass z \RR\)

\(=\)

\(\ds \eqclass {x \circ y} \RR \circ_{S / \RR} \eqclass z \RR\)

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

\(\ds \)

\(=\)

\(\ds \eqclass {\paren {x \circ y} \circ z} \RR\)

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

\(\ds \)

\(=\)

\(\ds \eqclass {x \circ \paren {y \circ z} } \RR\)

Semigroup Axiom $\text S 1$: Associativity

\(\ds \)

\(=\)

\(\ds \eqclass x \RR \circ_{S / \RR} \eqclass {y \circ z} \RR\)

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

\(\ds \)

\(=\)

\(\ds \eqclass x \RR \circ_{S / \RR} \paren {\eqclass y \RR \circ_{S / \RR} \eqclass z \RR}\)

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

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

$\blacksquare$