Power-associative loop
This is a variation of group|Find other variations of group | Read a survey article on varying group
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 | |FULL LIST, MORE INFO | |||
| Moufang loop | |FULL LIST, MORE INFO | |||
| Left Bol loop | |FULL LIST, MORE INFO | |||
| Left Bruck 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 |