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