Complete embeddings of linear orderings and embeddings of lattice-ordered groups
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.
KeywordsAutomorphism Group Linear Ordering Regular Cardinal Tense Logic Infinite Cardinal
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- 2.M. Droste,Structure of partially ordered sets with transitive automorphism groups, Memoirs Amer. Math. Soc.334 (1985).Google Scholar
- 4.M. Droste,Completeness properties of certain normal subgroup lattices, European J. Combinatorics, to appear.Google Scholar
- 5.M. Droste,Partially ordered sets with transitive automorphism groups, Proc. London Math. Soc., to appear.Google Scholar
- 7.A. M. W. Glass,Ordered Permutation Groups, London Math. Soc. Lecture Note Series, Vol. 55, Cambridge, 1981.Google Scholar
- 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
- 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
- 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
- 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