Definition:Cancellable Element/Right Cancellable

Definition

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

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

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

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

• Definition:Left Cancellable Element
• Definition:Cancellable Element

• Definition:Right Cancellable Operation

• Definition:Cancellable Operation
• Definition:Left Cancellable Operation

• 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 $7.6$