Lattice-ordered groups whose lattices determine their additions

Conrad, Paul F.; Darnel, Michael R.

doi:10.1090/S0002-9947-1992-1031238-0

Lattice-ordered groups whose lattices determine their additions
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by Paul F. Conrad and Michael R. Darnel PDF

Trans. Amer. Math. Soc. 330 (1992), 575-598 Request permission

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

Groupes et anneaux réticulés

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

Lattice-ordered groups

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

groups with a unique addition

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

Lattice-ordered groups

Partially ordered algebraic structures

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

Die Axiome der Quantität und die Lehre vom

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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