# Definition:Quotient Structure

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## Definition

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 $S / \RR$.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures