Power-associative loop

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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.

Relation with other structures

Stronger structures

• Group
• Moufang loop
• Alternative loop

Weaker structures

• Algebra loop
• Quasigroup