Definition:Well-Defined/Operation

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Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\mathcal R$ be a congruence for $\circ$.

Let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$.


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

$\eqclass x {\mathcal R} \circ_\mathcal R \eqclass y {\mathcal R} = \eqclass {x \circ y} {\mathcal R}$


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

$x, x' \in \eqclass x {\mathcal R}, y, y' \in \eqclass y {\mathcal R} \implies x \circ y = x' \circ y'$


Also known as

Some sources use the term consistent for well-defined.


Sources