# Definition:Left Cancellable Mapping

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## Definition

A mapping $f: Y \to Z$ is **left cancellable** (or **left-cancellable**) if and only if:

- $\forall X: \forall \struct {g_1, g_2: X \to Y}: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$

That is, for any set $X$, if $g_1$ and $g_2$ are mappings from $X$ to $Y$:

- If $f \circ g_1 = f \circ g_2$
- then $g_1 = g_2$.

## Also known as

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

Hence the property of **being cancellable** is given 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 left cancellable mapping can be considered as a left 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): $\S 1.14$: Exercise $19 \ \text{(a)}$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.5$: Identity, One-one, and Onto Functions