# Definition:Left Cancellable Mapping

## 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

Some text call this a left cancellative mapping.

## 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.