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Lattice-ordered Permutation Groups: The Structure Theory

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Ordered Groups and Infinite Permutation Groups

Part of the book series: Mathematics and Its Applications ((MAIA,volume 354))

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Abstract

This survey of lattice-ordered permutation groups focuses especially on their structure theory and on their relation to unordered infinite permutation groups. However, the survey is essentially self-contained. The reader will need only a little familiarity with unordered permutation groups, and none at all with lattice-ordered groups.

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References

  1. R. N. Ball, Full convex â„“-subgroups of a lattice-ordered group, Ph.D. Thesis, University of Wisconsin, Madison, Wis., USA, 1974.

    Google Scholar 

  2. R. N. Ball and M. Droste, Normal subgroups of doubly transitive automorphism groups of chains, Trans. Amer. Math. Soc. 290 (1985), 647–664.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. R. Darnel, M. Giraudet, and S. H. McCleary, Uniqueness of the group operation on the lattice of order-automorphisms of the real line, Algebra Universalis 33 (1995), 419–427.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Droste, Representations of free lattice-ordered groups, Order 10(1993), 375– 381.

    Google Scholar 

  5. M. Droste and S. H. McCleary, The root system of prime subgroups of a free lattice-ordered group (without GCH), Order 6 (1989), 305–309.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Droste and S. Shelah, A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets, Israel J. Math. 51 (1985), 223–261.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. M. W. Glass, L-simple lattice-ordered groups, Proc. Edinburgh Math. Soc. 19 (1974), 133–138.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. M. W. Glass, Ordered Permutation Groups, London Math. Soc. Lecture Notes Series No. 55, Cambridge University Press, Cambridge, England, 1981.

    Google Scholar 

  9. A. M. W. Glass and W. C. Holland (editors), Lattice-ordered groups: Advances and Techniques, Kluwer Academic Pub., Dordrecht, The Netherlands, 1989.

    MATH  Google Scholar 

  10. A. M. W. Glass and S. H. McCleary, Big subgroups of automorphism groups of doubly homogeneous chains, Ordered Algebraic Structures: The 1991 Con-rad Conference, ed. J. Martinez and C. Holland, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, 51–71.

    Google Scholar 

  11. C. Gourion, A propos du groupe des automorphismes de (Q;≤), C. R. Acad. Sci. Paris Sér. I Math. 315(1992), 1329–1331.

    MathSciNet  MATH  Google Scholar 

  12. Yu. Gurevich and W. C. Holland, Recognizing the real line, Trans. Amer. Math. Soc. 265 (1981), 527–534.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Higman, On infinite simple groups, Publ. Math. Debrecen 3 (1954), 221–226.

    MathSciNet  MATH  Google Scholar 

  14. W. C. Holland, The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10 (1963), 399–408.

    Article  MathSciNet  MATH  Google Scholar 

  15. W. C. Holland, Transitive lattice-ordered permutation groups, Math. Zeit. 87 (1965), 420–433.

    Article  MathSciNet  MATH  Google Scholar 

  16. W. C. Holland,A class of simple lattice-ordered permutation groups, Proc. Amer. Math. Soc. 16 (1965), 326–329.

    Article  MathSciNet  MATH  Google Scholar 

  17. W. C. Holland,Outer automorphisms of ordered permutation groups, Proc. Edinburgh Math. Soc. 19 (1975), 331–344.

    Article  MathSciNet  MATH  Google Scholar 

  18. W. C. Holland, The largest proper variety of lattice-ordered groups, Proc. Amer. Math. Soc. 57 (1976), 25–28.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. C. Holland, Partial orders of the group of automorphisms of the real line, Contemporary Math. 131 (1992), 197–207.

    MathSciNet  Google Scholar 

  20. W. C. Holland and S. H. McCleary, Wreath products of ordered permutation groups, Pacific J. Math. 31 (1969), 703–716.

    MathSciNet  MATH  Google Scholar 

  21. W. C. Holland, Solvability of the word problem in free lattice-ordered groups, Houston J. Math. 5 (1979), 99–105.

    MathSciNet  MATH  Google Scholar 

  22. M. (Jambu-)Giraudet, Bi-interpretable groups and lattices, Trans. Amer. Math. Soc. 278(1983), 253–269.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Kneser, Kurvenscharen auf Ringflächen, Math. Ann. 91 (1924), 135–154.

    Article  MathSciNet  MATH  Google Scholar 

  24. V. M. Kopytov, Free lattice-ordered groups, Algebra and Logic 18(1979), 259–270 (English translation).

    Article  MathSciNet  Google Scholar 

  25. J. T. Lloyd, Lattice-ordered groups and o-permutation groups, Ph.D. Thesis, Tulane University, New Orleans, La., USA, 1964.

    Google Scholar 

  26. J. T. Lloyd, Complete distributivity in certain infinite permutation groups, Michigan Math. J. 14 (1967), 393–400.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. D. Macpherson, Large subgroups of infinite symmetric groups, Proc. NATO AST Conf. on Finite and Infinite Combinatorics, Banff, Alberta, Canada, 1991.

    Google Scholar 

  28. S. H. McCleary, Orbit configurations of ordered permutation groups, Ph.D. Thesis, University of Wisconsin, Madison, Wis., USA, 1967.

    Google Scholar 

  29. S. H. McCleary, The closed prime subgroups of certain ordered permutation groups, Pacific J. Math. 31 (1969), 745–753.

    MathSciNet  MATH  Google Scholar 

  30. S. H. McCleary, O-primitive ordered permutation groups, Pacific J. Math. 40 (1972), 349–372.

    MathSciNet  MATH  Google Scholar 

  31. S. H. McCleary, Closed subgroups of lattice-ordered permutation groups, Trans. Amer.Math. Soc. 173 (1972), 303–314.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. H. McCleary, 0-2-transitive ordered permutation groups, Pacific J. Math. 49 (1973), 425–429.

    MathSciNet  MATH  Google Scholar 

  33. S. H. McCleary, O-primitive ordered permutation groups, Pacific J. Math. 49 (1973), 431–443.

    MathSciNet  MATH  Google Scholar 

  34. S. H. McCleary, The structure of intransitive ordered permutation groups, Algebra Universalis 6 (1976), 229–255.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. H. McCleary, Free lattice-ordered groups represented as o-2-transitive ℓ-permutation groups, Trans. Amer. Math. Soc. 290 (1985), 69–79.

    MathSciNet  MATH  Google Scholar 

  36. S. H. McCleary, An even better representation for free lattice-ordered groups, Trans. Amer. Math. Soc. 290 (1985), 81–100.

    Article  MathSciNet  MATH  Google Scholar 

  37. S. H. McCleary and M. Rubin, Locally moving groups and the reconstruction problem for chains and circles, in preparation.

    Google Scholar 

  38. T. Ohkuma, Sur quelques ensembles ordonnés linéairement, Fund. Math. 43 (1955), 326–337.

    MathSciNet  Google Scholar 

  39. J. A. Read, Wreath products of nonoverlapping lattice-ordered groups, Canad. Math. Bull. 17 (1975), 713–722.

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Schreier and S. Ulam, Eine Bemerkung über die Gruppe der topologischen Abbildungen der Kreislinie auf sich selbst, Studia Math. 5 (1935), 155–159.

    MATH  Google Scholar 

  41. J. K. Truss, Lnfinite permutation groups: subgroups of small index, J. Algebra 120 (1989), 495–515.

    Google Scholar 

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© 1996 Kluwer Academic Publishers

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McCleary, S.H. (1996). Lattice-ordered Permutation Groups: The Structure Theory. In: Holland, W.C. (eds) Ordered Groups and Infinite Permutation Groups. Mathematics and Its Applications, vol 354. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3443-9_2

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  • DOI: https://doi.org/10.1007/978-1-4613-3443-9_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3445-3

  • Online ISBN: 978-1-4613-3443-9

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