# Definition:Cancellable Element

## Definition

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

An element $x \in \struct {S, \circ}$ 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, if and only if it is both left cancellable and right cancellable.

### 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 $x \in \struct {S, \circ}$ is **right cancellable** if and only if:

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

## 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

- Cancellable Elements of Semigroup form Subsemigroup
- Definition:Cancellable Operation
- Definition:Left Cancellable Operation
- Definition:Right Cancellable Operation

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.

- Results about
**cancellativity**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups