Cyclically ordered group

Definition
A cyclically ordered group is a set $$G$$ equipped with two structures:


 * The structure of a group
 * The structure of a cyclically ordered set

such that both left and right multiplication maps by group elements preserve the cyclic ordering.

The notion of cyclically ordered group generalizes the notion of linearly ordered group, because every linear ordering is a cyclic ordering. Further, cyclic orderings can be imposed on finite cyclic groups, whereas linear orderings can only be imposed on torsion-free groups.

The notion can further be generalized to that of partially cyclically ordered group.