Definition:Quotient Structure

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

Let $$\mathcal R$$ be a congruence for $$\circ$$, and let $$\circ_{\mathcal R}$$ be the operation induced on $S / \mathcal R$ by $\circ$.

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

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 $$S / \mathcal R$$.