---
tag: 6CR9A0EWPN
kind: concept
name: Cancellation property
msc: [08A02, 20M10]
sources:
  - url: https://en.wikipedia.org/wiki/Cancellation_property
same_as:
  wikidata: Q2298524
  wikipedia: Cancellation property
---

## Summary

An element $a$ has the cancellation property for a [[binary operation|0M418FE7BB]] when it can be removed from both sides of an equation $a \ast b = a \ast c$ or $b \ast a = c \ast a$.
A structure in which every element is cancellable is called cancellative.
Every [[group|G5S1M9]] is cancellative by the [[cancellation laws|0VSNX4P2Q0]], and the nonzero elements of an [[integral domain|0QVD7WBFGN]] are cancellable for multiplication.
Cancellative commutative [[semigroups|0XMEM406JM]] are exactly those that embed in a group.

## Explanations

### L1 human

Cancelling means: from $a + b = a + c$, conclude $b = c$.
This works for addition of natural numbers even though there is no subtraction inside the natural numbers.
It fails for multiplication by $0$: $0 \cdot 1 = 0 \cdot 2$, yet $1 \ne 2$.
So $0$ is not cancellable for multiplication, while every nonzero integer is.