Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence/Corollary

Corollary to Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence
Let $\struct {A, \oplus}$ be an algebraic structure.

Let $\RR$ and $\TT$ be congruence relations on $\struct {A, \oplus}$ such that $R \subseteq T$.

Let $\SS$ be the relation on the quotient structure $\struct {A / \RR, \oplus_\RR}$ which satisfies:
 * $\forall X, Y \in A / \RR: X \mathrel \SS Y \iff \exists x \in X, y \in Y: x \mathrel \TT y$

Then:
 * $\SS$ is a congruence relation on $\struct {A / \RR, \oplus_\RR}$

and:
 * there exists a unique isomorphism $\phi$ from $\paren {A / \RR} / \SS$ to $A / \TT$ which satisfies:
 * $\phi \circ q_\SS \circ q_\RR = q_\TT$
 * where $q_\SS$, $q_\RR$ and $q_\TT$ denote the quotient epimorphisms as appropriate.

Proof
Recall that by definition $\RR$ and $\TT$ are equivalence relations.

First it is demonstrated that $\SS$ is an equivalence relation.

Checking in turn each of the criteria for equivalence:

Reflexivity
Let $X \in A / \RR$.

Then as $\TT$ is an equivalence relation, therefore reflexive:
 * $\forall x \in X: x \mathrel \TT x$

Thus by definition of $\SS$:
 * $\forall X \in A / \RR: X \mathrel \SS X$

Thus $\SS$ is seen to be reflexive.

Symmetry
Thus $\SS$ is seen to be symmetric.

Transitivity
Let $X, Y, Z \in A / \RR$ such that

Then we have:

Thus $\SS$ is seen to be transitive.

Hence $\TT$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

It remains to be demonstrated that $\SS$ is a congruence relation.

Let $X_1, Y_1, X_2, Y_2 \in A / \RR$ such that:

Then we have:

Hence by definition of congruence relation:
 * $\SS$ is a congruence relation on $\struct {A / \RR, \oplus_\RR}$.

Thus we have that:


 * $\RR$ is a congruence relation on $\struct {A, \oplus}$

and:
 * $\SS$ is a congruence relation on the quotient structure $\struct {A / \RR, \oplus_\RR}$ defined by $\RR$.

By definition of quotient structure:


 * $\forall x, y \in A: x \mathrel \TT y \iff \eqclass x \RR \mathrel \SS \eqclass y \RR$

where $\eqclass x \RR$ is the equivalence class under $\RR$ of $x$.

Hence the criteria of Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence are fulfilled.

Hence from Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence:
 * there exists a unique isomorphism $\phi$ from $\paren {A / \RR} / \SS$ to $A / \TT$ which satisfies:
 * $\phi \circ q_\SS \circ q_\RR = q_\TT$
 * where $q_\SS$, $q_\RR$ and $q_\TT$ denote the quotient epimorphisms as appropriate.