# Definition:Cancellable Element

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## 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

- 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.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 7$