# 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 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.