Definition:Quotient Structure

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

• Quotient Structure is Well-Defined

• Results about quotient structures can be found here.

Linguistic Note

The word quotient derives from the Latin word meaning how often.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): quotient: 3.