Power-associative loop

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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 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