Cancellation property

Definition

Let $\ast$ be a binary operation on a set $S$ and let $a \in S$ .

The element $a$ is left cancellable if $a \ast b = a \ast c$ implies $b = c$ , for all $b, c \in S$ .
The element $a$ is right cancellable if $b \ast a = c \ast a$ implies $b = c$ , for all $b, c \in S$ .
The element $a$ is cancellable if it is both left and right cancellable.

The pair $(S, \ast)$ is left cancellative, right cancellative or cancellative if every element of $S$ is left cancellable, right cancellable or cancellable.

Remarks

Example: $(\N, +)$ is cancellative, because $a + b = a + c$ gives $b = c$ by comparing the numbers, and $+$ is commutative.

Non-example: $(\Z, \cdot)$ is not cancellative, because $0 \cdot 1 = 0 \cdot 2$ with $1 \ne 2$ , so $0$ is not left cancellable.

An element $a$ is left cancellable if and only if the map $x \mapsto a \ast x$ is injective. On a set with at least two elements, the operation $a \ast b = b$ is left cancellative but not right cancellative.

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Binary operation concept
Cancellation property definition

Tag	`2JVB5R0MBW`
URI	`m2://2JVB5R0MBW`
Kind	definition
Writing	W0 stub
MSC	08A02, 20M10
Defines	Cancellation property
Uses	Binary operation

Cancellation property

Definition

Remarks

Reading path

Further reading

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