Definition:Cancellable Operation
Definition
Let $\struct {S, \circ}$ be an algebraic structure.
The operation $\circ$ in $\struct {S, \circ}$ is cancellable if and only if:
- $\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$
- $\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$
That is, if and only if it is both a left cancellable operation and a right cancellable operation.
Left Cancellable Operation
The operation $\circ$ in $\struct {S, \circ}$ is left cancellable if and only if:
- $\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$
That is, if and only if all elements of $\struct {S, \circ}$ are left cancellable.
Right Cancellable Operation
The operation $\circ$ in $\struct {S, \circ}$ is right cancellable if and only if:
- $\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$
That is, if and only if all elements of $\struct {S, \circ}$ are right cancellable.
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.