# Magma in which cubes and fourth powers are well-defined

From Groupprops

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.

View other such properties

## Contents

## Definition

A magma is termed a **magma in which cubes and fourth powers are well-defined** if it satisfies the following conditions for all (here, is shorthand for ):

- Cubes are well-defined: , and the common value is denoted .
- Fourth powers are well-defined: , and the common value is denoted .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Magma in which powers up to the fifth are well-defined | fifth powers are also well-defined | |FULL LIST, MORE INFO | ||

Alternative magma | satisfies left-alternative and right-alternative laws | via alternative implies powers up to the fifth are well-defined | Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO | |

Power-associative magma | all positive powers are well-defined | Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO | ||

Jordan magma | Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Magma in which cubes are well-defined | cubes well-defined, fourth powers need not be | Magma in which cubes are well-defined and every element commutes with its cube|FULL LIST, MORE INFO | ||

Magma in which cubes are well-defined and every element commutes with its cube | cubes well-defined, every element commutes with its cube | |FULL LIST, MORE INFO |