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Cyclically ordered group

Summary

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 edit

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 edit

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 edit

Every compact cyclically ordered group is a subgroup of T.

Related structures edit

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 edit

  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 edit

  • 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 edit

  • 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
  • Černák, Štefan (1989b), "Cantor extension of an Abelian cyclically ordered group" (PDF), Mathematica Slovaca, 39 (1): 31–41, hdl:10338.dmlcz/128948, retrieved 21 May 2011
  • Černák, Štefan (1991), "On the completion of cyclically ordered groups" (PDF), Mathematica Slovaca, 41 (1): 41–49, hdl:10338.dmlcz/131783, retrieved 22 May 2011
  • Černák, Štefan (1995), "Lexicographic products of cyclically ordered groups" (PDF), Mathematica Slovaca, 45 (1): 29–38, hdl:10338.dmlcz/130473, retrieved 21 May 2011
  • 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
  • Černák, Štefan (2002), "Completion of a half linearly cyclically ordered group", Discussiones Mathematicae - General Algebra and Applications, 22 (1): 5–23, doi:10.7151/dmgaa.1043
  • Černák, Štefan; Jakubík, Ján (1987), "Completion of a cyclically ordered group", Czechoslovak Mathematical Journal, 37 (1): 157–174, doi:10.21136/CMJ.1987.102144, hdl:10338.dmlcz/102144, MR 0875137, Zbl 0624.06021
  • Fuchs, László (1963), "IV.6. Cyclically ordered groups", Partially ordered algebraic systems, International series of monographs in pure and applied mathematics, vol. 28, Pergamon Press, pp. 61–65, LCC QA171 .F82 1963
  • Giraudet, M.; Kuhlmann, F.-V.; Leloup, G. (February 2005), "Formal power series with cyclically ordered exponents" (PDF), Archiv der Mathematik, 84 (2): 118–130, CiteSeerX 10.1.1.6.5601, doi:10.1007/s00013-004-1145-5, S2CID 16156556, retrieved 30 April 2011
  • Harminc, Matúš (1988), "Sequential convergences on cyclically ordered groups" (PDF), Mathematica Slovaca, 38 (3): 249–253, hdl:10338.dmlcz/128594, retrieved 21 May 2011
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  • Jakubík, Ján (1991), "Completions and closures of cyclically ordered groups" (PDF), Czechoslovak Mathematical Journal, 41 (1): 160–169, doi:10.21136/CMJ.1991.102447, hdl:10338.dmlcz/102447, MR 1087637, retrieved 21 May 2011
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  • Jakubík, Ján (2008), "Sequential convergences on cyclically ordered groups without Urysohn's axiom", Mathematica Slovaca, 58 (6): 739–754, doi:10.2478/s12175-008-0105-0
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