\begin{definition}[Definition:Quotient Structure]
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\RR$ be a congruence relation on $\struct {S, \circ}$.
Let $S / \RR$ be the quotient set of $S$ by $\RR$.
Let $\circ_\RR$ be the operation induced on $S / \RR$ by $\circ$.
The '''quotient structure defined by $\RR$''' is the algebraic structure:
:$\struct {S / \RR, \circ_\RR}$
If there is no danger of confusion, we can drop the $\RR$ from $\circ_\RR$ and use $\circ$ for both the operation on $S$ and the induced operation on $S / \RR$.
\end{definition}