Definition:Well-Defined/Operation

Definition

Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\mathcal R$ be a congruence for $\circ$, and let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$.

Let $\left({S / \mathcal R, \circ_\mathcal R}\right)$ be the quotient structure defined by $\mathcal R$, where $\circ_\mathcal R$ is defined as:

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

Then $\circ_\mathcal R$ is well-defined (on $S / \mathcal R$) iff:

$x, x' \in \left[\!\left[{x}\right]\!\right]_\mathcal R, y, y' \in \left[\!\left[{y}\right]\!\right]_\mathcal R \implies x \circ y = x' \circ y'$

Also known as

Some sources use the term consistent for well-defined.