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 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.
Generalizations[]
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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.
- ^ a b c 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" (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: , 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[]
- 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" (PDF), Discussiones Mathematicae - General Algebra and Applications, 21 (1): 31–46, doi:10.7151/dmgaa.1025, retrieved 22 May 2011[permanent dead link]
- Černák, Štefan (2002), "Completion of a half linearly cyclically ordered group" (PDF), Discussiones Mathematicae - General Algebra and Applications, 22 (1): 5–23, doi:10.7151/dmgaa.1043, retrieved 22 May 2011[permanent dead link]
- 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
- 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" (PDF), Archivum Mathematicum, 25 (1): 13–18, hdl:10338.dmlcz/107334, 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, hdl:10338.dmlcz/102406
- 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
- Jakubík, Ján (1998), "Lexicographic product decompositions of cyclically ordered groups" (PDF), Czechoslovak Mathematical Journal, 48 (2): 229–241, doi:10.1023/A:1022881202595, hdl:10338.dmlcz/127413, S2CID 55134686, retrieved 21 May 2011
- Jakubík, Ján (2002), "On half cyclically ordered groups" (PDF), Czechoslovak Mathematical Journal, 52 (2): 275–294, doi:10.1023/A:1021718426347, hdl:10338.dmlcz/127716, S2CID 117967332, 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" (PDF), Časopis Pro Pěstování Matematiky, 113 (2): 184–196, doi:10.21136/CPM.1988.118342, hdl:10338.dmlcz/118342, retrieved 30 April 2011
- Jakubík, Ján; Pringerová, Gabriela (1988), "Radical classes of cyclically ordered groups" (PDF), Mathematica Slovaca, 38 (3): 255–268, hdl:10338.dmlcz/129356, retrieved 30 April 2011
- Jakubík, Ján; Pringerová, Gabriela (1994), "Direct limits of cyclically ordered groups" (PDF), Czechoslovak Mathematical Journal, 44 (2): 231–250, doi:10.21136/CMJ.1994.128465, hdl:10338.dmlcz/128465, retrieved 21 May 2011
- Leloup, Gérard (2007), "Cyclically valued rings and formal power series", Annales Mathématiques Blaise Pascal, 14 (1): 37–60, doi:10.5802/ambp.226, retrieved 30 April 2011
- Lenz, Hanfried (1967), "Zur Begründung der Winkelmessung", Mathematische Nachrichten, 33 (5–6): 363–375, doi:10.1002/mana.19670330510
- Luce, R. Duncan (1971), "Periodic extensive measurement", Compositio Mathematica, 23 (2): 189–198, retrieved 22 May 2011
- Oltikar, B. C. (March 1980). "Right cyclically ordered groups". Canadian Mathematical Bulletin. 23 (1): 67–70. doi:10.4153/CMB-1980-009-3. MR 0573560.
- Pecinová, Eliška (2008), Ladislav Svante Rieger (1916–1963), Dějiny matematiky (in Czech), vol. 36, Prague: Matfyzpress, hdl:10338.dmlcz/400757, ISBN 978-80-7378-047-0, retrieved 9 May 2011
- Rieger, L. S. (1946), "О uspořádaných a cyklicky uspořádaných grupách I (On ordered and cyclically ordered groups I)", Věstník Královské české Spolecnosti Nauk, Třída Mathematicko-přírodovědná (Journal of the Royal Czech Society of Sciences, Mathematics and Natural History) (in Czech) (6): 1–31
- Rieger, L. S. (1947), "О uspořádaných a cyklicky uspořádaných grupách II (On ordered and cyclically ordered groups II)", Věstník Královské české Spolecnosti Nauk, Třída Mathematicko-přírodovědná (Journal of the Royal Czech Society of Sciences, Mathematics and Natural History) (in Czech) (1): 1–33
- Rieger, L. S. (1948), "О uspořádaných a cyklicky uspořádaných grupách III (On ordered and cyclically ordered groups III)", Věstník Královské české Spolecnosti Nauk, Třída Mathematicko-přírodovědná (Journal of the Royal Czech Society of Sciences, Mathematics and Natural History) (in Czech) (1): 1–22
- Roll, J. Blair (1976), On manipold groups: a generalization of the concept of cyclically ordered groups, Bowling Green State University, OCLC 3193754
- Roll, J. Blair (1993), "Locally partially ordered groups" (PDF), Czechoslovak Mathematical Journal, 43 (3): 467–481, doi:10.21136/CMJ.1993.128411, hdl:10338.dmlcz/128411, retrieved 30 April 2011
- Vinogradov, A. A. (1970), "Ordered algebraic systems", in Filippov, N. D. (ed.), Ten Papers on algebra and functional analysis, American Mathematical Society Translations, Series 2, vol. 96, AMS Bookstore, pp. 69–118, ISBN 978-0-8218-1796-4
- Walker, Harold Allen (1972), Cyclically ordered semigroups (Thesis), University of Tennessee, OCLC 54363006
- Zabarina, Anna Ivanovna (1982), "Theory of cyclically ordered groups", Mathematical Notes, 31 (1): 3–8, doi:10.1007/BF01146259, S2CID 121833530. Translation of Zabarina (1982), К теории циклически упорядоченных групп, Matematicheskie Zametki (in Russian), 31 (1): 3–12, retrieved 22 May 2011
- Zabarina, Anna Ivanovna (1985), "Linear and cyclic orders in a group", Sibirskii Matematicheskii Zhurnal (in Russian), 26 (2): 204–207, 225, MR 0788349
- Zabarina, Anna Ivanovna; Pestov, German Gavrilovich (1984), "Sverchkovskii's theorem", Siberian Mathematical Journal, 24 (4): 545–551, doi:10.1007/BF00968891, S2CID 121613711. Translation from Sibirskii Matematicheskii Zhurnal, 46–53
- Zabarina, Anna Ivanovna; Pestov, German Gavrilovich (1986), "On a criterion for cyclic orderability of a group", Uporyadochennye Mnozhestva I Reshetki (in Russian), 9: 19–24, Zbl 0713.20034
- Zassenhaus, Hans (June–July 1954), "What is an Angle?", The American Mathematical Monthly, 61 (6): 369–378, doi:10.2307/2307896, JSTOR 2307896
- Želeva, S. D. (1976), "On cyclically ordered groups", Sibirskii Matematicheskii Zhurnal (in Russian), 17: 1046–1051, MR 0422106, Zbl 0362.06022
- Želeva, S. D. (1981), "Half-homogeneously cyclically ordered groups", Godishnik Vyssh. Uchebn. Zaved. Prilozhna Mat. (in Russian), 17 (4): 123–126, MR 0705070, Zbl 0511.06013
- Želeva, S. D. (1981), "Cyclically and T-like ordered groups", Godishnik Vyssh. Uchebn. Zaved. Prilozhna Mat. (in Russian), 17 (4): 137–149, MR 0705071, Zbl 0511.06014
- Želeva, S. D. (1985), "A group of automorphisms of a cyclically ordered set", Nauchni Tr., Plovdivski Univ., Mat. (in Bulgarian), 23 (2): 25–31, Zbl 0636.06009
- Želeva, S. D. (1985), "A partial right ordering of the group of automorphisms of a cyclically ordered set", Nauchni Tr., Plovdivski Univ., Mat. (in Bulgarian), 23 (2): 47–56, Zbl 0636.06011
- Želeva, S. D. (1997), "Representation of right cyclically ordered groups as groups of automorphisms of a cyclically ordered set", Mathematica Balkanica, New Series, 11 (3–4): 291–294, Zbl 1036.06501
- Želeva, S. D. (1998), "Lattice cyclically ordered groups", Mathematica Balkanica, New Series, 12 (1–2): 47–58, Zbl 1036.06502
- Ordered groups
- Circles