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.[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
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
(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
Every compact cyclically ordered group is a subgroup of T.
Related structures
(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
- ↑ Pecinová-Kozáková 2005, p. 194.
- ↑ Świerczkowski 1959a, p. 162.
- ↑ 3.0 3.1 3.2 Świerczkowski 1959a, pp. 161–162.
- ↑ Hölder 1901, cited after Hofmann & Lawson 1996, pp. 19, 21, 37
- ↑ Gluschankof 1993, p. 261.
References
- Gluschankof, Daniel (1993), "Cyclic ordered groups and MV-algebras", Czechoslovak Mathematical Journal 43 (2): 249–263, doi:10.21136/CMJ.1993.128391, http://dml.cz/bitstream/handle/10338.dmlcz/128391/CzechMathJ_43-1993-2_6.pdf, 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 978-0-521-57669-7
- 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 978-80-86732-59-6
- Świerczkowski, S. (1959a), "On cyclically ordered groups", Fundamenta Mathematicae 47 (2): 161–166, doi:10.4064/fm-47-2-161-166, http://matwbn.icm.edu.pl/ksiazki/fm/fm47/fm4718.pdf, retrieved 2 May 2011
Further reading
- Černák, Štefan (1989a), "Completion and Cantor extension of cyclically ordered groups", in Hałkowska, Katarzyna; Stawski, Boguslaw, Universal and Applied Algebra (Turawa, 1988), World Scientific, pp. 13–22, ISBN 978-9971-5-0837-1
- Černák, Štefan (1989b), "Cantor extension of an Abelian cyclically ordered group", Mathematica Slovaca 39 (1): 31–41, http://www.dml.cz/bitstream/handle/10338.dmlcz/128948/MathSlov_39-1989-1_6.pdf, retrieved 21 May 2011
- Černák, Štefan (1991), "On the completion of cyclically ordered groups", Mathematica Slovaca 41 (1): 41–49, http://www.dml.cz/bitstream/handle/10338.dmlcz/131783/MathSlov_41-1991-1_7.pdf, retrieved 22 May 2011
- Černák, Štefan (1995), "Lexicographic products of cyclically ordered groups", Mathematica Slovaca 45 (1): 29–38, http://dml.cz/bitstream/handle/10338.dmlcz/130473/MathSlov_45-1995-1_4.pdf, retrieved 21 May 2011
- Černák, Štefan; Jakubík, Ján (1987), "Completion of a cyclically ordered group", Czechoslovak Mathematical Journal 37 (1): 157–174, doi:10.21136/CMJ.1987.102144
- Fuchs, László (1963), "IV.6. Cyclically ordered groups", Partially ordered algebraic systems, International series of monographs in pure and applied mathematics, 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", Archiv der Mathematik 84 (2): 118–130, doi:10.1007/s00013-004-1145-5, http://math.usask.ca/~fvk/gklcor.pdf, retrieved 30 April 2011
- Harminc, Matúš (1988), "Sequential convergences on cyclically ordered groups", Mathematica Slovaca 38 (3): 249–253, http://dml.cz/bitstream/handle/10338.dmlcz/128594/MathSlov_38-1988-3_8.pdf, retrieved 21 May 2011
- Hölder, O. (1901), "Die Axiome der Quantität und die Lehre vom Mass", Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physicke Klasse 53: 1–64
- Jakubík, Ján (1989), "Retracts of abelian cyclically ordered groups", Archivum Mathematicum 25 (1): 13–18, http://dml.cz/bitstream/handle/10338.dmlcz/107334/ArchMath_025-1989-1_3.pdf, retrieved 21 May 2011
- Jakubík, Ján (1990), "Cyclically ordered groups with unique addition", Czechoslovak Mathematical Journal 40 (3): 534–538, doi:10.21136/CMJ.1990.102406
- Jakubík, Ján (1991), "Completions and closures of cyclically ordered groups", Czechoslovak Mathematical Journal 41 (1): 160–169, doi:10.21136/CMJ.1991.102447, http://dml.cz/bitstream/handle/10338.dmlcz/102447/CzechMathJ_41-1991-1_21.pdf, retrieved 21 May 2011
- Jakubík, Ján (1998), "Lexicographic product decompositions of cyclically ordered groups", Czechoslovak Mathematical Journal 48 (2): 229–241, doi:10.1023/A:1022881202595, http://dml.cz/bitstream/handle/10338.dmlcz/127413/CzechMathJ_48-1998-2_3.pdf, retrieved 21 May 2011
- Jakubík, Ján (2002), "On half cyclically ordered groups", Czechoslovak Mathematical Journal 52 (2): 275–294, doi:10.1023/A:1021718426347, http://dml.cz/bitstream/handle/10338.dmlcz/127716/CzechMathJ_52-2002-2_4.pdf, retrieved 22 May 2011
- 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
- Jakubík, Ján; Pringerová, Gabriela (1988), "Representations of cyclically ordered groups", Časopis Pro Pěstování Matematiky 113 (2): 184–196, doi:10.21136/CPM.1988.118342, http://dml.cz/bitstream/handle/10338.dmlcz/118342/CasPestMat_113-1988-2_6.pdf, retrieved 30 April 2011
- Jakubík, Ján; Pringerová, Gabriela (1988), "Radical classes of cyclically ordered groups", Mathematica Slovaca 38 (3): 255–268, http://dml.cz/bitstream/handle/10338.dmlcz/129356/MathSlov_38-1988-3_9.pdf, retrieved 30 April 2011
- Jakubík, Ján; Pringerová, Gabriela (1994), "Direct limits of cyclically ordered groups", Czechoslovak Mathematical Journal 44 (2): 231–250, doi:10.21136/CMJ.1994.128465, http://dml.cz/bitstream/handle/10338.dmlcz/128465/CzechMathJ_44-1994-2_3.pdf, retrieved 21 May 2011
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