Cyclically ordered group

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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[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.

Generalizations[edit]

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", Czechoslovak Mathematical Journal 43 (2): 249–263, retrieved 30 April 2011 
  • Hofmann, Karl H.; Lawson, Jimmie D. (1996), "A survey on totally ordered semigroups", in Hofmann, Karl H.; Mislove, Michael W., Semigroup theory and its applications: proceedings of the 1994 conference commemorating the work of Alfred H. Clifford, London Mathematical Society Lecture Note Series 231, Cambridge University Press, pp. 15–39, ISBN 0-521-57669-5 
  • Pecinová-Kozáková, Eliška (2005), "Ladislav Svante Rieger and His Algebraic Work", in Safrankova, Jana, WDS 2005 - Proceedings of Contributed Papers, Part I, Prague: Matfyzpress, pp. 190–197, ISBN 80-86732-59-2, retrieved 25 April 2011 
  • Świerczkowski, S. (1959a), "On cyclically ordered groups", Fundamenta Mathematicae 47: 161–166, retrieved 2 May 2011 

Further reading[edit]