Definition:Operation Induced on Quotient Set

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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 $\mathcal R$.


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

$\left[\!\left[{x}\right]\!\right]_\mathcal R \circ_\mathcal R \left[\!\left[{y}\right]\!\right]_\mathcal R = \left[\!\left[{x \circ y}\right]\!\right]_\mathcal R$


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