Cyclically ordered group

Template:Short description 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 Z² / Z. Rieger (1946, 1947, 1948) 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, xⁿ, 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, −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".

References

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Further reading

• Černák, Štefan (1989a), "Completion and Cantor extension of cyclically ordered groups", in Hałkowska, Katarzyna; Stawski, Boguslaw (eds.), Universal and Applied Algebra (Turawa, 1988), World Scientific, pp. 13–22, ISBN 978-9971-5-0837-1, MR 1084391
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• Černák, Štefan (2001), "Cantor extension of a half linearly cyclically ordered group", Discussiones Mathematicae - General Algebra and Applications, 21 (1): 31–46, doi:10.7151/dmgaa.1025
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