Definition:Operation Induced on Quotient Set

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 $\RR$.

The operation $\circ_\RR$ induced on $S / \RR$ by $\circ$ is defined as:

$\eqclass x \RR \circ_\RR \eqclass y \RR = \eqclass {x \circ y} \RR$

Also see

• Definition:Quotient Structure
• Quotient Structure is Well-Defined

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups