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.

Generalizations

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".