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 $Z$ and the finite cyclic groups $Z/n$. Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers $Q$, the real numbers $R$, and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group $T$ 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 $Z_{n} = Z/nZ$ and $T = R/Z$. Even a once-linear group like $Z$, when bent into a circle, can be thought of as $Z^{2} / Z$. showed that this picture is a generic phenomenon. For any ordered group $L$ and any central element $z$ that generates a cofinal subgroup $Z$ of $L$, the quotient group $L / Z$ 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 $K$ and an ordered group $L$, the product $K × L$ is a cyclically ordered group. In particular, if $T$ is the circle group and $L$ is an ordered group, then any subgroup of $T × L$ is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with $T$.

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 $x, y$ such that $[e, x^{n}, y]$ for every positive integer $n$. Since only positive $n$ are considered, this is a stronger condition than its linear counterpart. For example, $Z$ no longer qualifies, since one has $[0, n, &minus;1]$ for every $n$.

As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of $T$ itself. This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of $R$.

Topology
Every compact cyclically ordered group is a subgroup of $T$.

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".