Definition:Cancellable Element/Left Cancellable
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Definition
Let $\struct {S, \circ}$ be an algebraic structure.
An element $x \in \struct {S, \circ}$ is left cancellable if and only if:
- $\forall a, b \in S: x \circ a = x \circ b \implies a = b$
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
- 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 $7.6$