# Alternative magma

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 an **alternative magma** if it is both a left-alternative magma and a right-alternative magma, i.e., it satisfies the following two identities:

## Relation with other properties

### Property obtained by the opposite operation

Suppose is a magma and we define on as . Then, is an alternative magma if and only if is an alternative magma.

### Stronger properties

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

Diassociative magma | submagma generated by any two elements is associative | |FULL LIST, MORE INFO | ||

Semigroup | whole magma is associative | Diassociative magma|FULL LIST, MORE INFO |

### Weaker properties

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

Left-alternative magma | |FULL LIST, MORE INFO | |||

Right-alternative magma | |FULL LIST, MORE INFO | |||

Magma in which cubes are well-defined | Left alternative magma, Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined, Right-alternative magma|FULL LIST, MORE INFO | |||

Magma in which powers up to the fifth are well-defined | well-defined for all | alternative implies powers up to the fifth are well-defined | |FULL LIST, MORE INFO |