Definition:Renaming Mapping

Definition

Let $f: S \to T$ be a mapping.

The renaming mapping $r: S / \mathcal R_f \to \Img f$ is defined as:

$r: S / \mathcal R_f \to \Img f: \map r {\eqclass x {\mathcal R_f} } = \map f x$

where:

$\mathcal R_f$ is the equivalence induced by the mapping $f$

$S / \mathcal R_f$ is the quotient set of $S$ determined by $\mathcal R_f$

$\eqclass x {\mathcal R_f}$ is the equivalence class of $x$ under $\mathcal R_f$.

Also known as

This mapping can also be seen referred to as the mapping on $S / \mathcal R_f$ induced by $f$.

However, the term induced mapping is used so often throughout this area of mathematics that it would make sense to use a less-overused term whenever possible.

Also see

• Existence of Renaming Mapping
• Quotient Theorem for Sets
• Renaming Mapping is Well-Defined
• Renaming Mapping is Bijection

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts ... (previous) ... (next): Introduction $\S 3$: Equivalence relations
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{I}$: Factoring Functions
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.5$