Definition:Quotient Structure

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\mathcal R$ be a congruence relation on $\left({S, \circ}\right)$.

Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$.

Let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$.

The quotient structure defined by $\mathcal R$ is the algebraic structure:
 * $\left({S / \mathcal R, \circ_\mathcal R}\right)$

If there is no danger of confusion, we can drop the $\mathcal R$ from $\circ_\mathcal R$ and use $\circ$ for both the operation on $S$ and the induced operation on $S / \mathcal R$.

Also see

 * Quotient Structure is Well-Defined