Cyclically ordered group
In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947. They are a generalization of cyclic groups: the infinite cyclic group  and the finite cyclic groups . Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers , the real numbers , and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group  and its subgroups, such as the subgroup of rational points. Quotients of linear groups
It is natural to depict cyclically ordered groups as quotients: one has  and . Even a once-linear group like , when bent into a circle, can be thought of as . showed that this picture is a generic phenomenon. For any ordered group  and any central element  that generates a cofinal subgroup  of , the quotient group  is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as such a quotient group. The circle group
 built upon Rieger's results in another direction. Given a cyclically ordered group  and an ordered group , the product  is a cyclically ordered group. In particular, if  is the circle group and  is an ordered group, then any subgroup of  is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with . By analogy with an Archimedean linearly ordered group, one can define an Archimedean cyclically ordered group as a group that does not contain any pair of elements  such that  for every positive integer . Since only positive  are considered, this is a stronger condition than its linear counterpart. For example,  no longer qualifies, since one has  for every . As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of  itself. This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of . Topology
Every compact cyclically ordered group is a subgroup of . Related structures
 showed that a certain subcategory of cyclically ordered groups, the "projectable Ic-groups with weak unit", is equivalent to a certain subcategory of MV-algebras, the "projectable MV-algebras". Notes

References

Further reading

. Translation of 

. Translation from Sibirskii Matematicheskii Zhurnal, 46–53

Ordered groups
Circles