# Power-associative loop

From Groupprops

This is a variation of group|Find other variations of group | Read a survey article on varying group

## Contents

## Definition

A **power-associative loop** is an algebra loop satisfying the following equivalent conditions:

- The subloop generated by any element is a cyclic subgroup, i.e., it is associative.
- Every element is contained in a subgroup with the same identity element.

Note that this is somewhat *stronger* than simply being a power-associative magma because we care here about positive *and* negative powers.

## Relation with other properties

### Stronger properties

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

Group | Diassociative loop, Moufang loop|FULL LIST, MORE INFO | |||

Moufang loop | Diassociative loop, Moufang loop|FULL LIST, MORE INFO | |||

Left Bol loop | |FULL LIST, MORE INFO | |||

Left Bruck loop | Left Bol loop|FULL LIST, MORE INFO |

### Weaker properties

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

Loop | |FULL LIST, MORE INFO | |||

Quasigroup | |FULL LIST, MORE INFO | |||

Power-associative magma | |FULL LIST, MORE INFO |