# Definition:Cancellable Monoid

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## Definition

Let $\struct {S, \circ}$ be a monoid.

$\struct {S, \circ}$ is defined as being **cancellable** if and only if:

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

That is, if and only if $\circ$ is a cancellable operation.

## 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
**cancellable monoids**can be found**here**.

## Sources

- 1999: J.C. Rosales and P.A. García-Sánchez:
*Finitely Generated Commutative Monoids*... (previous) ... (next): Chapter $1$: Basic Definitions and Results