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On cyclically ordered groups. (Autour des groupes cycliquement ordonnés.) (French. English summary) Zbl 1270.06012

The notion of a cyclically ordered group was introduced by Rieger as a group \(C\) endowed with a ternary relation \(R\) satisfying the conditions:
(1)
\(R(a,b,c) \Rightarrow a\neq b\neq c\neq a,\)
(2)
\(R(a,b,c) \Rightarrow R(b,c,a),\)
(3)
for every \(c\in C\), \(R(c,\cdot,\cdot)\) defines a linearly ordered relation on \(C\setminus \{c\},\)
(4)
\(R(\cdot,\cdot,\cdot)\) is compatible, i.e., \(R(a,b,c)\Rightarrow R(uav,ubv,ucv)\),
for all \((a,b,c,u,v)\in C^5\).
The paper under review investigates special properties of cyclically ordered groups, such as: = 5mm
\(-\)
lexicographic product;
\(-\)
a group \(G\) is cycle-orderabe if and only its center \(Z(G)\) is orderable;
\(-\)
if \((C, R)\) is a cyclically ordered abelian group and \(H\) is a \(c\)-convex subgroup of \(C\), then \(C, R)\) is the lexcographic product of \((C/H)\) and \(H\);
\(-\)
properties regarding the minimality;
\(-\)
rings of formal power series with exponents in a cyclically ordered group;
\(-\)
cyclically valued groups;
\(-\)
cyclically ultrametric spaces.

MSC:

06F15 Ordered groups
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[1] Droste (M.), Giraudet (M.) & Macpherson (D.).— Periodic ordered permutation groups and cyclic orderings, Journal of Combinatorial Theory, Series B 63, p. 310-321 (1995). · Zbl 0821.20001
[2] Fuchs (L.).— Partially Ordered Algebraic Systems, International Series of Monographs in Pure and Applied Mathematics, vol 28, Pergamon Press (1963). · Zbl 0137.02001
[3] Giraudet (M.), Kuhlmann (F.-V.) & Leloup (G.).— Formal power series with cyclically ordered exponents, Arch. Math. 84, p. 118-130 (2005). · Zbl 1090.13015
[4] Giraudet (M.) & Lucas (F.).— Quelques résultats sur la théorie des modèles de groupes abéliens cycliquement ordonnés, Séminaire de Structures Algébriques ordonnées, prépublications de l’université Paris VII 49 (1994).
[5] Giraudet (M.) & Lucas (F.).— First order theory of cyclically ordered groups, dans : F. Lucas, Théorie des modèles de groupes abéliens ordonnés, habilitation à diriger des recherches, université Paris VII, (1996).
[6] Giraudet (M.) & Lucas (F.).— c-homomorphisms and c-convex subgroups of cyclically ordered groups, preprint. · Zbl 0766.06014
[7] Glusschankov (D.).— Cyclic ordered groups and MV-algebras, Czechoslovak Mathematikal Journal \(\textbf{43} n^{\circ } 2\), p. 249-263 (1983). · Zbl 0795.06015
[8] Jakubík (J.) & Pringerová (G.).— Representations of cyclically ordered groups. Časop. Pěstov. Matem. 113, p. 184-196 (1988). · Zbl 0654.06016
[9] Jakubík (J.) & Pringerová (G.).— Direct limits of cyclically ordered groups. Czechoslovak Mathematikal Journal \(\textbf{44} n^{\circ } 2\), p. 231-250 (1994). · Zbl 0821.06015
[10] Kaplansky (I.).— Maximal fields with valuations, Duke Math Journal 9, p. 303-321 (1942). · Zbl 0063.03135
[11] Kokorin (A.I.) & Kopytov (V.M.).— Fully ordered groups, J. Wiley & sons, New-York (1974).
[12] Leloup (G.).— Cyclically valued rings and formal power series, Annales mathématiques Blaise Pascal 14, \(n^{\circ } 1\), p. 117-140 (2007).
[13] Leloup (G.).— Existentially equivalent cyclically ultrametric spaces and cyclically valued groups, Logic Journal of the IGPL, 2010 ; doi :10.1093/jigpal/jz024. · Zbl 1216.03052
[14] Lucas (F.).— Théorie des modèles des groupes cycliquement ordonnés abéliens divisibles, dans : F. Lucas, Théorie des modèles de groupes abéliens ordonnés, habilitation à diriger des recherches, université Paris VII (1996).
[15] Lucas (F.).— Théorie des modèles des groupes cycliquement ordonnés abéliens divisibles, Séminaire de Structures Algébriques ordonnées, prépublications de l’université Paris VII 56 (1996).
[16] Novák (V.).— Cyclically ordered sets, Czech. Math. J. 32, p. 460-473 (1982). · Zbl 0515.06003
[17] Novák (V.).— Cuts in cyclically ordered sets, Czech. Math. J. 34, p. 322-333 (1984). · Zbl 0551.06002
[18] Novák (V.) & Novotný (M.).— On completions of cyclically ordered sets, Czech. Math. J. 37, p. 407-414 (1987). · Zbl 0636.06004
[19] Rieger (L.).— On ordered and cyclically ordered groups I, Věstník král. českŽ spol. nauk, p. 1-31 (en tchèque) (1946).
[20] Rieger (L.).— On ordered and cyclically ordered groups II, Věstník král. českŽ spol. nauk, p. 1-33 (en tchèque) (1947).
[21] Rieger (L.).— On ordered and cyclically ordered groups III, Věstník král. českŽ spol. nauk, p. 1-26 (en tchèque) (1948).
[22] Sabbagh (G.).— Un théorème de plongement en algèbre, Bull. Soc. Math. 2, p. 49-52 (1968). · Zbl 0159.33802
[23] Schmitt (P.-H.).— Undecidable theories of valuated Abelian groups, Mémoires de la S.M.F. \(2^{\mbox{ème}}\) série, tome 16, p. 67-76 (1984). · Zbl 0555.20034
[24] Schmitt (P.-H.).— Decidable theories of valuated Abelian groups, Proceedings of the Logic Colloquium 1984, North Holland, p. 245-276 (1986). · Zbl 0604.03012
[25] Świerczkowski (S.).— On cyclically ordered groups, Fund. Math. 47, p. 161-166 (1959). · Zbl 0096.01501
[26] Zabrina (I.) & Pestov (G.).— Sverchkovskii’s theorem, Sibirskii’s Mathematischeskii Zurnal \(\textbf{25} n^{\circ } 4\), p. 46-53 (1984). · Zbl 0554.06013
[27] Zheleva (S. D.).— Cyclically ordred groups, Sibirskii Mathematiskii Zhurnal 17, 1046-1051 (1976). · Zbl 0362.06022
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