Definition:Cancellable Element
Definition
Let $\struct {S, \circ}$ be an algebraic structure.
An element $x \in \struct {S, \circ}$ is cancellable if and only if:
- $\forall a, b \in S: x \circ a = x \circ b \implies a = b$
- $\forall a, b \in S: a \circ x = b \circ x \implies a = b$
That is, if and only if it is both left cancellable and right cancellable.
Left Cancellable
An element $x \in \struct {S, \circ}$ is left cancellable if and only if:
- $\forall a, b \in S: x \circ a = x \circ b \implies a = b$
Right Cancellable
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
- Cancellable Elements of Semigroup form Subsemigroup
- Definition:Cancellable Operation
- Definition:Left Cancellable Operation
- Definition:Right Cancellable Operation
In the context of mapping theory:
from which it can be seen that:
- a right cancellable mapping can be considered as a right cancellable element
- a left cancellable mapping can be considered as a left cancellable element
of an algebraic structure whose operation is composition of mappings.
- 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