# Definition:Cancellable Element

## Definition

Let $\left ({S, \circ}\right)$ be an algebraic structure.

### Left Cancellable

An element $x \in \struct {S, \circ}$ is left cancellable if and only if:

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

### Right Cancellable

An element $x \in \left ({S, \circ}\right)$ is right cancellable iff:

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

### Cancellable

An element $x \in \left ({S, \circ}\right)$ is cancellable if and only if:

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

... that is, it is both left cancellable and right cancellable.

## Also known as

Some authors use regular to mean cancellable, but this usage can be ambiguous so is not generally endorsed.

## Also see

In the context of mapping theory:

from which it can be seen that:

a right cancellable mapping can be considered as a right cancellable element
a left cancellable mapping can be considered as a left cancellable element

of an algebraic structure whose operation is composition of mappings.