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 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 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