Quotient Structure is Similar to Structure

Theorem

Let $\RR$ be a congruence relation on a algebraic structure $\struct {G, \circ}$.

Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a similar structure to $\struct {G, \circ}$.

Proof

Quotient Structure of Semigroup is Semigroup

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 $\circ$ $\ds$ $=$ $\ds \eqclass {x \circ \paren {y \circ z} } \RR$ $\circ$ is Associative $\ds$ $=$ $\ds \eqclass x \RR \circ_{S / \RR} \eqclass {y \circ z} \RR$ 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.

$\blacksquare$

Quotient Structure of Monoid is Monoid

From Quotient Structure of Semigroup is Semigroup $\struct {S / \RR, \circ_\RR}$ is a semigroup.

Let $\eqclass x {\RR} \in S / \RR$.

Consider $\eqclass e \RR$:

 $\ds \eqclass x \RR \circ_{S / \RR} \eqclass e \RR$ $=$ $\ds \eqclass {x \circ e} \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$ $\ds$ $=$ $\ds \eqclass x \RR$ Definition of Identity Element

Furthermore:

 $\ds \eqclass e \RR \circ_{S / \RR} \eqclass x \RR$ $=$ $\ds \eqclass {e \circ x} \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$ $\ds$ $=$ $\ds \eqclass x \RR$ Definition of Identity Element

Hence $\eqclass e \RR$ is an identity.

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

$\blacksquare$

Quotient Structure of Group is Group

From Quotient Structure of Monoid is Monoid $\struct {G / \RR, \circ_\RR}$ is a monoid with $\eqclass e \RR$ as its identity.

Let $\eqclass x \RR \in S / \RR$.

Consider $\eqclass {-x} \RR$ where $-x$ denotes the inverse of $x$ under $\circ$.

 $\ds \eqclass x \RR \circ_{S / \RR} \eqclass {-x} \RR$ $=$ $\ds \eqclass {x \circ -x} \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$ $\ds$ $=$ $\ds \eqclass e \RR$ Definition of Inverse Element

Furthermore:

 $\ds \eqclass {-x} \RR \circ_{S / \RR} \eqclass x \RR$ $=$ $\ds \eqclass {-x \circ x} \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$ $\ds$ $=$ $\ds \eqclass e \RR$ Definition of Inverse Element

Hence $\eqclass {-x} \RR$ is the inverse of $\eqclass x \RR$.

Hence $\struct {G / \RR, \circ_\RR}$ is a group.

$\blacksquare$

Quotient Structure of Abelian Group is Abelian Group

From Quotient Structure of Group is Group we have that $\struct {G / \RR, \circ_\RR}$ is a group.

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

 $\ds \eqclass x \RR \circ_{S / \RR} \eqclass y \RR$ $=$ $\ds \eqclass {x \circ y} \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$ $\ds$ $=$ $\ds \eqclass {y \circ x} \RR$ $\circ$ is Commutative $\ds$ $=$ $\ds \eqclass y \RR \circ_{S / \RR} \eqclass x \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$

Hence $\circ_{S / \RR}$ is commutative.

Hence $\struct {G / \RR, \circ_\RR}$ is an abelian group.

$\blacksquare$