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.
Preview
Unable to display preview. Download preview PDF.
References
R. N. Ball, Full convex â„“-subgroups of a lattice-ordered group, Ph.D. Thesis, University of Wisconsin, Madison, Wis., USA, 1974.
R. N. Ball and M. Droste, Normal subgroups of doubly transitive automorphism groups of chains, Trans. Amer. Math. Soc. 290 (1985), 647–664.
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.
M. Droste, Representations of free lattice-ordered groups, Order 10(1993), 375– 381.
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.
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.
A. M. W. Glass, L-simple lattice-ordered groups, Proc. Edinburgh Math. Soc. 19 (1974), 133–138.
A. M. W. Glass, Ordered Permutation Groups, London Math. Soc. Lecture Notes Series No. 55, Cambridge University Press, Cambridge, England, 1981.
A. M. W. Glass and W. C. Holland (editors), Lattice-ordered groups: Advances and Techniques, Kluwer Academic Pub., Dordrecht, The Netherlands, 1989.
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.
C. Gourion, A propos du groupe des automorphismes de (Q;≤), C. R. Acad. Sci. Paris Sér. I Math. 315(1992), 1329–1331.
Yu. Gurevich and W. C. Holland, Recognizing the real line, Trans. Amer. Math. Soc. 265 (1981), 527–534.
G. Higman, On infinite simple groups, Publ. Math. Debrecen 3 (1954), 221–226.
W. C. Holland, The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10 (1963), 399–408.
W. C. Holland, Transitive lattice-ordered permutation groups, Math. Zeit. 87 (1965), 420–433.
W. C. Holland,A class of simple lattice-ordered permutation groups, Proc. Amer. Math. Soc. 16 (1965), 326–329.
W. C. Holland,Outer automorphisms of ordered permutation groups, Proc. Edinburgh Math. Soc. 19 (1975), 331–344.
W. C. Holland, The largest proper variety of lattice-ordered groups, Proc. Amer. Math. Soc. 57 (1976), 25–28.
W. C. Holland, Partial orders of the group of automorphisms of the real line, Contemporary Math. 131 (1992), 197–207.
W. C. Holland and S. H. McCleary, Wreath products of ordered permutation groups, Pacific J. Math. 31 (1969), 703–716.
W. C. Holland, Solvability of the word problem in free lattice-ordered groups, Houston J. Math. 5 (1979), 99–105.
M. (Jambu-)Giraudet, Bi-interpretable groups and lattices, Trans. Amer. Math. Soc. 278(1983), 253–269.
H. Kneser, Kurvenscharen auf Ringflächen, Math. Ann. 91 (1924), 135–154.
V. M. Kopytov, Free lattice-ordered groups, Algebra and Logic 18(1979), 259–270 (English translation).
J. T. Lloyd, Lattice-ordered groups and o-permutation groups, Ph.D. Thesis, Tulane University, New Orleans, La., USA, 1964.
J. T. Lloyd, Complete distributivity in certain infinite permutation groups, Michigan Math. J. 14 (1967), 393–400.
H. D. Macpherson, Large subgroups of infinite symmetric groups, Proc. NATO AST Conf. on Finite and Infinite Combinatorics, Banff, Alberta, Canada, 1991.
S. H. McCleary, Orbit configurations of ordered permutation groups, Ph.D. Thesis, University of Wisconsin, Madison, Wis., USA, 1967.
S. H. McCleary, The closed prime subgroups of certain ordered permutation groups, Pacific J. Math. 31 (1969), 745–753.
S. H. McCleary, O-primitive ordered permutation groups, Pacific J. Math. 40 (1972), 349–372.
S. H. McCleary, Closed subgroups of lattice-ordered permutation groups, Trans. Amer.Math. Soc. 173 (1972), 303–314.
S. H. McCleary, 0-2-transitive ordered permutation groups, Pacific J. Math. 49 (1973), 425–429.
S. H. McCleary, O-primitive ordered permutation groups, Pacific J. Math. 49 (1973), 431–443.
S. H. McCleary, The structure of intransitive ordered permutation groups, Algebra Universalis 6 (1976), 229–255.
S. H. McCleary, Free lattice-ordered groups represented as o-2-transitive ℓ-permutation groups, Trans. Amer. Math. Soc. 290 (1985), 69–79.
S. H. McCleary, An even better representation for free lattice-ordered groups, Trans. Amer. Math. Soc. 290 (1985), 81–100.
S. H. McCleary and M. Rubin, Locally moving groups and the reconstruction problem for chains and circles, in preparation.
T. Ohkuma, Sur quelques ensembles ordonnés linéairement, Fund. Math. 43 (1955), 326–337.
J. A. Read, Wreath products of nonoverlapping lattice-ordered groups, Canad. Math. Bull. 17 (1975), 713–722.
J. Schreier and S. Ulam, Eine Bemerkung über die Gruppe der topologischen Abbildungen der Kreislinie auf sich selbst, Studia Math. 5 (1935), 155–159.
J. K. Truss, Lnfinite permutation groups: subgroups of small index, J. Algebra 120 (1989), 495–515.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
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
Download citation
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
eBook Packages: Springer Book Archive