Abstract
We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically ‘nice’ properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S i ) of infinite linearly ordered sets (S i , ≤) such that each group A(S i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. N. Ball and M. Droste (1985) Normal subgroups of doubly transitive automorphism groups of chains, Trans. Amer. Math. Soc. 290, 647–664.
G. Birkhoff (1973) Lattice Theory, 3rd edn., Amer. Math. Soc. Colloq. Publ., vol. 25, Providence, Rhode Island.
M. Droste (1985) The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets, Order 2, 291–319.
M. Droste (1987) Completeness properties of certain normal subgroup lattices, Europ. J. Combinatorics 8, 129–137.
M. Droste (1986) Complete embeddings of linear orderings and embeddings of lattice-ordered groups, Israel J. Math. 56, 315–334.
M. Droste and S. Shelah (1985) A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets, Israel J. Math. 51, 223–261.
A. M. W. Glass (1981) Ordered Permutation Groups, London Math. Soc. Lecture Note Series, vol. 55, Cambridge.
A. M. W. Glass, Y. Gurevich, W. C. Holland, and M. Jambu-Giraudet (1981) Elementary theory of automorphism groups of doubly homogeneous chains, in Logic Year 1979–80, The University of Connecticut (eds. M. Lerman, J. H. Schmerl and R. I. Soare), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, pp. 67–82.
A. M. W. Glass and K. R. Pierce (1980) Existentially complete lattice-ordered groups, Israel J. Math. 36, 257–272.
Y. Gurevich and W. C. Holland (1981) Recognizing the real line, Trans. Amer. Math. Soc. 265, 527–534.
G. Higman (1954) On infinite simple permutation groups, Publ. Math. Debrecen 3, 221–226.
W. C. Holland (1963) The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10, 399–408.
W. C. Holland (1976) The largest proper variety of lattice-ordered groups, Proc. Amer. Math. Soc. 57, 25–28.
M. Jambu-Giraudet (1983) Bi-interpretable groups and lattices, Trans. Amer. Math. Soc. 278, 253–269.
M. Jambu-Giraudet (1985) Quelques remarques sur l'equivalence elementaire entre groupes ou treillis d'automorphismes de chaines 2-homogenes, Discrete Math. 53, 117–124.
T. Jech (1978) Set Theory, Pure and Applied Mathematics, Academic Press, New York.
J. T. Lloyd (1964) Lattice-ordered groups and o-permutation groups, PhD Thesis, Tulane University, New Orleans, Louisiana, U.S.A.
S. H. McCleary (1973) The lattice-ordered group of automorphisms of an α-set, Pacific J. Math. 49, 417–424.
S. H. McCleary (1978) Groups of homeomorphisms with manageable automorphism groups, Comm. Algebra 6, 497–528.
G. F. McNulty (1980) Classes which generate the variety of all lattice-ordered groups, in Ordered Groups, Proc. Boise State Conference (eds. J. E. Smith, G. O. Kenny and R. N. Ball), Lecture Notes in Pure and Applied Mathematics, vol. 62, Marcel Dekker, New York, pp. 135–140.
E. C. Weinberg (1980) Automorphism groups of minimal n α -sets, in Ordered Groups, Proc. Boise State Conference (eds. J. E. Smith, G. O. Kenny and R. N. Ball), Lecture Notes in Pure and Applied Mathematics, vol. 62, Marcel Dekker, New York, pp. 71–79.
Author information
Authors and Affiliations
Additional information
Communicated by A. M. W. Glass
Rights and permissions
About this article
Cite this article
Droste, M. Normal subgroups and elementary theories of lattice-ordered groups. Order 5, 261–273 (1988). https://doi.org/10.1007/BF00354894
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00354894