Definition:Cancellable Element

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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 if and only if:

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

... 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:

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