Definition:Cancellable Element

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

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


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

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


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

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 and right cancellable.

Some authors use regular to mean cancellable, but this usage can be ambiguous so is not generally endorsed.