# Category:Cancellability

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This category contains results about **cancellable elements**.

Definitions specific to this category can be found in Definitions/Cancellability.

Let $\struct {S, \circ}$ be an algebraic structure.

An element $x \in \struct {S, \circ}$ is **cancellable** if and only if:

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

## Subcategories

This category has the following 7 subcategories, out of 7 total.

## Pages in category "Cancellability"

The following 33 pages are in this category, out of 33 total.

### C

- Cancellable Element is Cancellable in Subset
- Cancellable Elements of Semigroup form Subsemigroup
- Cancellable Finite Semigroup is Group
- Cancellable iff Regular Representations Injective
- Cancellable Infinite Semigroup is not necessarily Group
- Cancellation Laws
- Condition for Invertibility in Power Structure on Associative or Cancellable Operation

### I

- Identity is only Idempotent Cancellable Element
- Identity of Monoid is Cancellable
- Injection iff Left Cancellable
- Invertible Element of Associative Structure is Cancellable
- Invertible Elements of Monoid form Subgroup of Cancellable Elements
- Isomorphism Preserves Cancellability
- Isomorphism Preserves Left Cancellability
- Isomorphism Preserves Right Cancellability

### L

- Left Cancellable Commutative Operation is Right Cancellable
- Left Cancellable Elements of Semigroup form Subsemigroup
- Left Cancellable iff Left Regular Representation Injective
- Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection
- Lexicographically Ordered Pair of Ordered Semigroups with Cancellable Elements

### O

### P

### R

- Reflexive Reduction of Relation Compatible with Cancellable Operation is Compatible
- Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection
- Right Cancellable Commutative Operation is Left Cancellable
- Right Cancellable Elements of Semigroup form Subsemigroup
- Right Cancellable iff Right Regular Representation Injective
- Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection