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Lattice-ordered groups whose lattices determine their additions


Authors: Paul F. Conrad and Michael R. Darnel
Journal: Trans. Amer. Math. Soc. 330 (1992), 575-598
MSC: Primary 06F20
DOI: https://doi.org/10.1090/S0002-9947-1992-1031238-0
MathSciNet review: 1031238
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Abstract: In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or determined up to $ l$-isomorphism.


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  • [1] R. Ball, P. Conrad, and M. Darnel, Above and below subgroups of a lattice-ordered group, Trans. Amer. Math. Soc. 297 (1987), 1-40. MR 849464 (88b:06027)
  • [2] S. Bernau, Unique representation of lattice groups and normal archimedean lattice rings, Proc. London Math. Soc. 15 (1965), 599-631. MR 0182661 (32:144)
  • [3] A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math., vol. 608, Springer, 1979.
  • [4] J. P. Bixler and M. Darnel, Special-valued $ l$-groups, Algebra Universalis 22 (1986), 172-191. MR 870466 (88h:06021)
  • [5] R. Byrd, P. Conrad, and J. Lloyd, Characteristic subgroups of lattice-ordered groups, Trans. Amer. Math. Soc. 158 (1971), 339-371. MR 0279014 (43:4740)
  • [6] P. Conrad, Lattice-ordered groups, Tulane Lecture Notes, Tulane Univ., 1970.
  • [7] -, The essential closure of an archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151-160. MR 0277457 (43:3190)
  • [8] -, Epi-archimedean groups, Czechoslovak Math. J. 24 (1974), 1-27. MR 0347701 (50:203)
  • [9] P. Conrad and M. Darnel, $ l$-groups with a unique addition, Algebra and Order, Research and Exposition in Math. 14, Heldermann, 1986, pp. 19-27.
  • [10] P. Conrad and J. Diem, The ring of polar-preserving endomorphisms of an abelian lattice-ordered group, Illinois J. Math. 15 (1971), 222-240. MR 0285462 (44:2680)
  • [11] P. Conrad and D. McAlister, The completion of a lattice-ordered group, J. Austral. Math. Soc. 9 (1969), 182-208. MR 0249340 (40:2585)
  • [12] P. Conrad and P. McCarthy, The structure of $ f$-algebras, Math. Nachr. 58 (1973), 169-191. MR 0330000 (48:8339)
  • [13] M. Darnel, Lattice-ordered groups, Thesis, Univ. of Kansas, 1983.
  • [14] L. Fuchs, Partially ordered algebraic structures, Pergamon Press, 1963.
  • [15] A. M. W. Glass, Ordered permutation groups, London Math. Soc. Lecture Notes Series 55, Cambridge Univ. Press, 1981. MR 645351 (83j:06004)
  • [16] A. M. W. Glass, Y. Gurevich, W. C. Holland, and S. Shelah, Rigid homogeneous chains, Math. Proc. Cambridge Philos. Soc. 89 (1981), 7-17. MR 591966 (82c:06001)
  • [17] Ja. Hion, Archimedean ordered rings, Uspekhi Mat. Nauk 9:4 (1954), 237-242. MR 0065542 (16:442a)
  • [18] O. Hölder, Die Axiome der Quantität und die Lehre vom $ M\alpha \beta $, Ber. Verh. Sachs. Wiss. Leipzig, Math-Phys. Cl. 53 (1901), 1-64.
  • [19] W. C. Holland, Transitive lattice ordered permutation groups, Math. Z. 87 (1965), 420-433. MR 0178052 (31:2310)
  • [20] J. Jakubik, Representations and extensions of $ l$-groups, Czechoslovak Math. J. 13 (1963), 267-283. MR 0171865 (30:2091)
  • [21] M. Jambu-Giraudet, Bi-interpretable groups and lattices, Trans. Amer. Math. Soc. 278 (1983), 253-269. MR 697073 (84g:06028)
  • [22] L. Lavis, Sur les quotients totalements ordonnés d'un groupe linéairement ordonné, Bull. Soc. Royal Sci. Liege 32 (1963), 204-208. MR 0147565 (26:5080)
  • [23] C. Nobeling, Verallgemeinerung einer Satzes von Hern E. Specker, Invent. Math. 6 (1968), 41-55. MR 0231907 (38:233)
  • [24] T. Ohkuma, Sur quelques ensembles ordonnés linéairement, Fund. Math. 43 (1955), 326-337. MR 0084486 (18:868c)
  • [25] F. Sik, Zur theory den halbgeordneten gruppen, Czechoslovak Math. J. 6 (1956), 1-25. MR 0081907 (18:465a)
  • [26] E. Weinberg, Free lattice-ordered groups. II, Math. Ann. 159 (1965), 217-222. MR 0181668 (31:5895)
  • [27] S. Wolfenstein, Sur les groupes reitculés archimédiennement complets, C.R. Acad. Sci. Paris 262 (1966), A813-A816. MR 0194529 (33:2739)

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DOI: https://doi.org/10.1090/S0002-9947-1992-1031238-0
Article copyright: © Copyright 1992 American Mathematical Society

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