Definition:Cancellable Monoid

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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.