# Definition:Cancellable Element/Left Cancellable

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## Definition

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

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

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

## 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**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.6$