Definition:Well-Defined/Operation
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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