Israel Journal of Mathematics

, Volume 56, Issue 3, pp 315–334 | Cite as

Complete embeddings of linear orderings and embeddings of lattice-ordered groups

  • Manfred Droste


An infinite linearly ordered set (S,≦) is called doubly homogeneous, if its automorphism group Aut(S,≦) acts 2-transitively on it. We study embeddings of linearly ordered sets into Dedekind-completions of doubly homogeneous chains which preserve all suprema and infima, and obtain necessary and sufficient conditions for the existence of such embeddings. As one of several consequences, for each lattice-ordered groupG and each regular uncountable cardinalκ≧|G | there are 2⋉ non-isomorphic simple divisible lattice-ordered groupsH of cardinalityκ all containingG as anl-subgroup.


Automorphism Group Linear Ordering Regular Cardinal Tense Logic Infinite Cardinal 
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  1. 1.
    R. N. Ball and M. Droste,Normal subgroups of doubly transitive automorphism groups of chains, Trans. Amer. Math. Soc.290 (1985), 647–664.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Droste,Structure of partially ordered sets with transitive automorphism groups, Memoirs Amer. Math. Soc.334 (1985).Google Scholar
  3. 3.
    M. Droste,The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets, Order2 (1985), 291–319.MATHMathSciNetGoogle Scholar
  4. 4.
    M. Droste,Completeness properties of certain normal subgroup lattices, European J. Combinatorics, to appear.Google Scholar
  5. 5.
    M. Droste,Partially ordered sets with transitive automorphism groups, Proc. London Math. Soc., to appear.Google Scholar
  6. 6.
    M. Droste and S. Shelah,A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets, Isr. J. Math.51 (1985), 223–261.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. M. W. Glass,Ordered Permutation Groups, London Math. Soc. Lecture Note Series, Vol. 55, Cambridge, 1981.Google Scholar
  8. 8.
    G. Higman,On infinite simple permutation groups, Publ. Math. Debrecen3 (1954), 221–226.MathSciNetGoogle Scholar
  9. 9.
    W. C. Holland,The lattice-ordered group of automorphisms of an ordered set. Michigan Math. J.10 (1963), 399–408.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    W. C. Holland,Transitive lattice-ordered permutation groups, Math. Z.87 (1965), 420–433.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Jambu-Giraudet,Bi-interpretable groups and lattices, Trans. Amer. Math. Soc.278 (1983), 253–269.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. T. Lloyd,Lattice-ordered groups and o-permutation groups, Ph.D. Thesis, Tulane University, New Orleans, Louisiana, U.S.A., 1964.Google Scholar
  13. 13.
    S. H. McCleary,The closed prime subgroups of certain ordered permutation groups, Pacific J. Math.31 (1969), 745–753.MATHMathSciNetGoogle Scholar
  14. 14.
    S. H. McClearly,The lattice-ordered group of automorphisms of an α-set, Pacific J. Math.49 (1973), 417–424.MathSciNetGoogle Scholar
  15. 15.
    S. H. McCleary,Groups of homeomorphisms with manageable automorphism groups, Comm. Algebra6 (1978), 497–528.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    E. B. Rabinovich,On linearly ordered sets with 2-transitive groups of automorphisms, Vesti Akad. Navuk BSSR (Seria Fizika-Mate. Navuk)6 (1975), 10–17 (in Russian).Google Scholar
  17. 17.
    J. Rosenstein,Linear Orderings, Pure and applied mathematics, Academic Press, New York, 1982.MATHGoogle Scholar
  18. 18.
    R. M. Solovay,Real-valued measurable cardinals, inAxiomatic Set Theory (D. Scott, ed.), Proc. Symp. Pure Math.13 I, Amer. Math. Soc., Providence, 1971, pp. 397–428.Google Scholar
  19. 19.
    J. F. A. K. van Benthem,The Logic of Time, Reidel, Dordrecht, 1983.MATHGoogle Scholar
  20. 20.
    E. C. Weinberg,Embedding in a divisible lattice-ordered group, J. London Math. Soc.42 (1967), 504–506.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    E. C. Weinberg,Automorphism groups of minimal η a-sets, inOrdered Groups (J. E. Smith, G. O. Kenny and R. N. Ball, eds.), Proc. of the Boise State Conference, Lecture Notes in Pure and Applied Math., No. 62, Marcel Dekker, New York, 1980, pp. 71–79.Google Scholar
  22. 22.
    J. P. Burgess,Beyond tense logic, J. Philos. Logic13 (1984), 235–248.CrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • Manfred Droste
    • 1
  1. 1.Fachbereich 6 — MathematikUniversität GHS EssenEssen 1West Germany

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