Definition:Cancellable Monoid

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