# Definition:Cancellable Operation

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

The operation $\circ$ in $\struct {S, \circ}$ is cancellable if and only if:

$\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$
$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$

That is, if and only if it is both a left cancellable operation and a right cancellable operation.

### Left Cancellable Operation

The operation $\circ$ in $\struct {S, \circ}$ is left cancellable if and only if:

$\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$

That is, if and only if all elements of $\struct {S, \circ}$ are left cancellable.

### Right Cancellable Operation

The operation $\circ$ in $\struct {S, \circ}$ is right cancellable if and only if:

$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$

That is, if and only if all elements of $\struct {S, \circ}$ are right cancellable.

## 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

• Results about cancellability can be found here.