Definition:Right Cancellable Mapping

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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 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 right cancellable mapping can be considered as a right cancellable element of an algebraic structure whose operation is composition of mappings.