Quotient Structure of Semigroup is Semigroup

From ProofWiki
Jump to navigation Jump to search

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.


$\blacksquare$