Definition:Quotient Structure

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 $$S / \mathcal{R}$$.