# 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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 11$