A magma with underlying set and binary operation is termed a gyrogroup if the following hold:
- Left identity and left inverse: There is an element such that is a left neutral element and every element has a left inverse with respect to . In other words:
and for all , there exists such that:
- Gyroassociativity: For any , there is a unique element such that:
- Gyroautomorphism: is a magma automorphism of for all . This is called the Thomas gyration, or gyroautomorphism, of .
- Weak loop property: If , then
Thus, the definition of left gyrogroup differs from that of gyrogroup, only in that the left loop property is relaced by the weak loop property.
Relation with other structures
Representations of left gyrogroups
By a representation of a left gyrogroup, we mean an expression of it as a left gyrotransversal of a subgroup.
- Involutory decomposition of groups into twisted subgroups and subgroups by Tuval Foguel and Abraham A. Ungar