Definition:Well-Defined/Operation

Definition

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

Let $\RR$ be a congruence for $\circ$.

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

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

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

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

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

Also known as

Some sources use the term consistent for well-defined.

Some sources do not hyphenate: well defined.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets