Definition:Cancellable Element

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

An element $x \in \struct {S, \circ}$ is cancellable :
 * $\forall a, b \in S: x \circ a = x \circ b \implies a = b$
 * $\forall a, b \in S: a \circ x = b \circ x \implies a = b$

That is, it is both left cancellable and right cancellable.

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:
 * Definition:Right Cancellable Mapping
 * Definition:Left Cancellable Mapping
 * 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.