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The closed subgroup problem for lattice-ordered groups. (English) Zbl 0682.06011
Kenoyer gave two examples proving that in the lattice of convex $$\ell$$- subgroups of a lattice-ordered group it was impossible to distinguish the closed convex $$\ell$$-subgroups. However, one of his examples was non- normal-valued. Here the same question is recast for normal-valued $$\ell$$- groups, and it remains open. The main theorem gives an interesting class of $$\ell$$-groups in which closed convex $$\ell$$-subgroups can be recognized in the lattice of convex $$\ell$$-subgroups.
Reviewer: J.Martinez

##### MSC:
 06F15 Ordered groups
Full Text:
##### References:
 [1] M. Anderson, P. Conrad andG. O. Kenny, Splitting properties in archimedean ?-groups. J. Austral. Math. Soc. (A)23, 247-256 (1977). · Zbl 0369.06012 [2] R. N. Ball, P. Conrad andM. Darnel, Above and below subgroups of a lattice-ordered group. Trans. Amer. Math. Soc.297, 1-40 (1986). · Zbl 0628.06013 [3] S. Bernau, Lateral and Dedekind completion of archimedean lattice groups. J. London Math. Soc.12, 320-322 (1976). · Zbl 0333.06008 [4] A.Bigard, K.Keimel et S.Wolfenstein, Groupes et Anneaux Réticulés. LNM608, Berlin-Heidelberg-New York 1977. [5] J. P. Bixler andM. Darnel, Special-valued ?-groups. Algebra Univ.22, 172-191 (1986). · Zbl 0597.06014 [6] R. D. Byrd,M-polars in lattice-ordered groups. Czech. Math. J.18, 230-239 (1968). · Zbl 0174.06004 [7] P. Conrad, The lattice of all convex ?-subgroups of a lattice-ordered group. Czech. Math. J.15, 101-123 (1965). · Zbl 0135.06301 [8] P. Conrad, A characterization of lattice-ordered groups by their convex ?-subgroups. J. Austral. Math. Soc.7, 145-159 (1967). · Zbl 0154.27001 [9] P. Conrad, Epi-archimedean groups. Czech. Math. J.24, 1-27 (1974). · Zbl 0319.06009 [10] P.Conrad and J.Martinez, Locally finite conditions on lattice-ordered groups. Submitted. · Zbl 0688.06011 [11] D. B. Kenoyer, Recognizability in the lattice of convex ?-subgroups of a lattice-ordered group. Czech. Math. J. (109)34, 411-416 (1984). · Zbl 0587.06006 [12] J. Martinez, The hyper-archimedean kernel sequence of a lattice-ordered group. Bull. Austral.Math. Soc.10, 337-350 (1974). · Zbl 0275.06026 [13] J. Martinez, Pairwise-splitting lattice-ordered groups. Czech. Math. J.27, 545-551 (1977). · Zbl 0378.06009 [14] F. ?ik, Zur Theorie der halbgeordneten Gruppen. Czech. Math. J.10, 400-424 (1960).
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