# Magma in which powers up to the fifth 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 powers up to the fifth are well-defined** if it satisfies the following three conditions for all (here, we denote by ):

- Cubes are well-defined, i.e., . The common value is denoted .
- Fourth powers are well-defined, i.e., . The common value is denoted .
- Fifth powers are well-defined, i.e., . The common value is denoted .

## Relation with other properties

### Stronger properties

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

Power-associative magma | all positive powers are well-defined | (obvious) | |FULL LIST, MORE INFO | |

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

Jordan magma | commutative, satisfies Jordan's identity | Jordan implies 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 and fourth powers are well-defined | cubes, fourth powers well-defined | (obvious) | |FULL LIST, MORE INFO | |

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