# Left 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 a **left-alternative magma** if it satisfies the following identity:

.

## Relation with other properties

### Property obtained by the opposite operation

If is a magma and we define on by , then is a left-alternative magma if and only if is a right-alternative magma.

### Stronger properties

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

Alternative magma | both left and right-alternative | |FULL LIST, MORE INFO | ||

Semigroup | associativity holds universally | Alternative magma, Diassociative magma|FULL LIST, MORE INFO | ||

Diassociative magma | submagma generated by any two elements is associative | Alternative magma|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 | every element commutes with its square | |FULL LIST, MORE INFO |

### Incomparable properties

Property | Meaning | Proof of one non-implication | Proof of other non-implication | Notions stronger than both | Notions weaker than both |
---|---|---|---|---|---|

Flexible magma | Diassociative magma, Left Bol magma with neutral element|FULL LIST, MORE INFO | Magma in which cubes are well-defined|FULL LIST, MORE INFO | |||

Right-alternative magma | Alternative magma, Diassociative magma|FULL LIST, MORE INFO | Magma in which cubes are well-defined|FULL LIST, MORE INFO | |||

Power-associative magma | all powers are well-defined | Diassociative magma|FULL LIST, MORE INFO | Magma in which cubes are well-defined|FULL LIST, MORE INFO |