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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.[1] 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

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It is natural to depict cyclically ordered groups as quotients: one has Zn = 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.[2]

The circle group

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Świerczkowski (1959a) 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.[3]

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.[3] 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.[3] This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of R.[4]

Topology

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Every compact cyclically ordered group is a subgroup of T.

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Gluschankof (1993) 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".[5]

Notes

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  1. ^ Pecinová-Kozáková 2005, p. 194.
  2. ^ Świerczkowski 1959a, p. 162.
  3. ^ a b c Świerczkowski 1959a, pp. 161–162.
  4. ^ Hölder 1901, cited after Hofmann & Lawson 1996, pp. 19, 21, 37
  5. ^ Gluschankof 1993, p. 261.

References

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  • Gluschankof, Daniel (1993), "Cyclic ordered groups and MV-algebras" (PDF), Czechoslovak Mathematical Journal, 43 (2): 249–263, doi:10.21136/CMJ.1993.128391, retrieved 30 April 2011
  • Hofmann, Karl H.; Lawson, Jimmie D. (1996), "A survey on totally ordered semigroups", in Hofmann, Karl H.; Mislove, Michael W. (eds.), Semigroup theory and its applications: proceedings of the 1994 conference commemorating the work of Alfred H. Clifford, London Mathematical Society Lecture Note Series, vol. 231, Cambridge University Press, pp. 15–39, ISBN 978-0-521-57669-7
  • Pecinová-Kozáková, Eliška (2005), "Ladislav Svante Rieger and His Algebraic Work", in Safrankova, Jana (ed.), WDS 2005 - Proceedings of Contributed Papers, Part I, Prague: Matfyzpress, pp. 190–197, CiteSeerX 10.1.1.90.2398, ISBN 978-80-86732-59-6
  • Świerczkowski, S. (1959a), "On cyclically ordered groups" (PDF), Fundamenta Mathematicae, 47 (2): 161–166, doi:10.4064/fm-47-2-161-166, retrieved 2 May 2011

Further reading

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📚 Artikel Terkait di Wikipedia

Cyclic group

a cyclically ordered group is cyclic. A metacyclic group is a group containing a cyclic normal subgroup whose quotient is also cyclic. These groups include

Cyclic order

requirement results in a partial cyclic order. A set with a cyclic order is called a cyclically ordered set or simply a cycle.[nb] Some familiar cycles are discrete

Linearly ordered group

announced in 2020. Cyclically ordered group Hahn embedding theorem Partially ordered group Deroin, Navas & Rivas 2014, 1.1.1. For abelian groups see Def. 6 in

Partially ordered group

integrally closed. Cyclically ordered group – Group with a cyclic order respected by the group operation Linearly ordered group – Group with translationally

Ordered field

orderings. Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic

Free-by-cyclic group

In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group G {\displaystyle

Primary cyclic group

finitely generated abelian groups. The subgroups of a primary cyclic group are linearly ordered by inclusion. The only other groups that have this property

Ordered vector space

In mathematics, an ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with