Definition:Cancellable Element

Definition
Let $\left ({S, \circ}\right)$ be an algebraic structure.

Cancellable
An element $x \in \left ({S, \circ}\right)$ 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 known as
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:
 * 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.