Definition:Cancellable Element

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

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

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

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

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

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


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

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

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