Lattice-ordered groups whose lattices determine their additions
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- by Paul F. Conrad and Michael R. Darnel
- Trans. Amer. Math. Soc. 330 (1992), 575-598
- DOI: https://doi.org/10.1090/S0002-9947-1992-1031238-0
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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.References
- Richard N. Ball, Paul Conrad, and Michael Darnel, Above and below subgroups of a lattice-ordered group, Trans. Amer. Math. Soc. 297 (1986), no. 1, 1–40. MR 849464, DOI 10.1090/S0002-9947-1986-0849464-7
- S. J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599–631. MR 182661, DOI 10.1112/plms/s3-15.1.599 A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math., vol. 608, Springer, 1979.
- J. Patrick Bixler and Michael Darnel, Special-valued $l$-groups, Algebra Universalis 22 (1986), no. 2-3, 172–191. MR 870466, DOI 10.1007/BF01224024
- Richard D. Byrd, Paul Conrad, and Justin T. Lloyd, Characteristic subgroups of lattice-ordered groups, Trans. Amer. Math. Soc. 158 (1971), 339–371. MR 279014, DOI 10.1090/S0002-9947-1971-0279014-7 P. Conrad, Lattice-ordered groups, Tulane Lecture Notes, Tulane Univ., 1970.
- Paul Conrad, The essential closure of an Archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151–160. MR 277457
- Paul Conrad, Epi-archimedean groups, Czechoslovak Math. J. 24(99) (1974), 192–218. MR 347701 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.
- P. F. Conrad and J. E. Diem, The ring of polar preserving endomorphisms of an abelian lattice-ordered group, Illinois J. Math. 15 (1971), 222–240. MR 285462
- Paul Conrad and Donald McAlister, The completion of a lattice ordered group, J. Austral. Math. Soc. 9 (1969), 182–208. MR 0249340
- Paul Conrad and Paul McCarthy, The structure of $f$-algebras, Math. Nachr. 58 (1973), 169–191. MR 330000, DOI 10.1002/mana.19730580111 M. Darnel, Lattice-ordered groups, Thesis, Univ. of Kansas, 1983. L. Fuchs, Partially ordered algebraic structures, Pergamon Press, 1963.
- A. M. W. Glass, Ordered permutation groups, London Mathematical Society Lecture Note Series, vol. 55, Cambridge University Press, Cambridge-New York, 1981. MR 645351
- A. M. W. Glass, Yuri Gurevich, W. Charles Holland, and Saharon Shelah, Rigid homogeneous chains, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 7–17. MR 591966, DOI 10.1017/S0305004100057881
- Ya. V. Hion, Archimedean ordered rings, Uspehi Mat. Nauk (N.S.) 9 (1954), no. 4(62), 237–242 (Russian). MR 0065542 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.
- Charles Holland, Transitive lattice-ordered permutation groups, Math. Z. 87 (1965), 420–433. MR 178052, DOI 10.1007/BF01111722
- Ján Jakubík, Representation and extension of $l$-groups, Czechoslovak Math. J. 13(88) (1963), 267–283 (Russian, with German summary). MR 171865
- M. Jambu-Giraudet, Bi-interpretable groups and lattices, Trans. Amer. Math. Soc. 278 (1983), no. 1, 253–269. MR 697073, DOI 10.1090/S0002-9947-1983-0697073-2
- A. Lavis, Sur les quotients totalement ordonnés d’un groupe linéairement ordonné, Bull. Soc. Roy. Sci. Liège 32 (1963), 204–208 (French). MR 147565
- G. Nöbeling, Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math. 6 (1968), 41–55 (German). MR 231907, DOI 10.1007/BF01389832
- Tadashi Ohkuma, Sur quelques ensembles ordonnés linéairement, Fund. Math. 43 (1956), 326–337 (French). MR 84486
- František Šik, Zur Theorie der halbgeordneten Gruppen, Czechoslovak Math. J. 6(81) (1956), 1–25 (Russian, with German summary). MR 81907
- Elliot Carl Weinberg, Free lattice-ordered abelian groups. II, Math. Ann. 159 (1965), 217–222. MR 181668, DOI 10.1007/BF01362439
- Samuel Wolfenstein, Sur les groupes réticulés archimédiennement complets, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A813–816 (French). MR 194529
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- 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