Definition:Left Cancellable Mapping

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


 * $\forall X: \forall \left({g_1, g_2: X \to Y}\right): f \circ g_1 = f \circ g_2 \implies g_1 = g_2$

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

 * Right cancellable mapping
 * Injection iff Left Cancellable

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.
 * Left cancellable element
 * Right cancellable element