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 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

In the context of mapping theory:

• Right cancellable mapping
• Left cancellable mapping

from which it can be seen that:

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.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$