Definition:Cancellable Element

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

Cancellable
An element $x \in \left ({S, \circ}\right)$ is cancellable iff:


 * $\forall a, b \in S: x \circ a = x \circ b \implies a = b$
 * $\forall a, b \in S: a \circ x = b \circ x \implies a = b$

... 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.