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.