# Definition:Right Cancellable Mapping

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## 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

In the context of abstract algebra:

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