Definition:Right Cancellable Mapping

Definition

A mapping $f: X \to Y$ is right cancellable (or right-cancellable) if and only if:

$\forall Z: \forall \paren {h_1, h_2: Y \to Z}: h_1 \circ f = h_2 \circ f \implies h_1 = h_2$

That is, if and only if for any set $Z$:

If $h_1$ and $h_2$ are mappings from $Y$ to $Z$

then $h_1 \circ f = h_2 \circ f$ implies $h_1 = h_2$.

Also known as

An object that is cancellable can also be referred to as cancellative.

Hence the property of being cancellable is also referred to on $\mathsf{Pr} \infty \mathsf{fWiki}$ as cancellativity.

Some authors use regular to mean cancellable, but this usage can be ambiguous so is not generally endorsed.

Also see

• Definition:Left Cancellable Mapping
• Surjection iff Right Cancellable

In the context of abstract algebra:

• Definition:Right Cancellable Element
• Definition:Left Cancellable Element

from which it can be seen that a right cancellable mapping can be considered as a right cancellable element of an algebraic structure whose operation is composition of mappings.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{BB}$: Categorical Matters
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.5$: Identity, One-one, and Onto Functions