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On some completeness properties for lattice ordered groups. (English) Zbl 0835.06019
From the author’s introduction: A nonempty subclass \(X\) of \({\mathcal G}\) (the class of all lattice ordered groups) is said to be a radical class if it is closed with respect to a) convex \(\ell\)-subgroups, and b) joins of convex \(\ell\)-subgroups. A radical class which is closed with respect to homomorphic images is said to be a torsion class. Let \(G\) be an \(\ell\)-group. We shall consider the following conditions for \(G\): \((p)\) \(G\) is projectable, \((\alpha)\) \(G\) is abelian, \((\alpha (1))\) each order bounded disjoint subset of \(G\) has a supremum, \((\alpha (1 \sigma))\) each countable order bounded disjoint subset of \(G\) has a supremum, \((\alpha (2))\) for every disjoint sequence \((f_n)\) in \(G\) such that \(f_n \to 0\) in order, the element \(\sup \{f_n\}\) exists, \((\alpha (3))\) whenever \((f_n)\) and \((g_n)\) are sequences in \(G\) with \(f_n \leq g_m\) for all \(m,n\) such that \(\inf (g_n - f_n) = 0\), then there exists \(h \in G\) such that \(f_n \leq h \leq g_n\) for all \(n\), \((\alpha (4))\) whenever \((f_n)\) and \((g_n)\) are sequences in \(G\) such that \(f_n \leq g_n\) for all \(n\), there exists \(h \in G\) such that \(f_n \leq h \leq g_n\) for all \(n\), \((\alpha (5))\) \(G\) is uniformly complete (i.e. for every \(e \in G^+\), each \(e\)-uniform Cauchy sequence has an \(e\)-uniform limit), \((\alpha (6))\) for every disjoint set \(\{ f_\lambda\}\) in \(G\) which is order bounded there exists an element \(g \in G^+\) such that \(g - f_\lambda \perp f_\lambda\) for all \(\lambda\). For \(i \in \{1,2, \dots, 6\}\) we denote by \({\mathcal G}_{(\alpha (i))}\) the class of all \(\ell\)-groups which satisfy the condition \(\alpha (i)\). Analogously for the indices \(\alpha\), \(p\) and \(\alpha (1 \sigma)\). It is proved that \({\mathcal G}_{(\alpha (1))}\), \({\mathcal G}_{(\alpha (1 \sigma))}\) and \({\mathcal G}_{(\alpha(2))}\) fail to be torsion classes. If \(i \in \{1, 2, 4\}\) then \({\mathcal G}_{(\alpha (i))}\) is a radical class. The questions whether \({\mathcal G}_{(\alpha (i)) }\) for \(i \in \{3,5,6\}\) are radical classes remain open. Some partial results in these directions are established; e.g., it is shown that the class of abelian \(\ell\)-groups belonging to \({\mathcal G}_{(\alpha (3))}\), the class of projectable \(\ell\)-groups belonging to \({\mathcal G}_{(\alpha (6))}\), \({\mathcal G}_{(\alpha (3))}\cap {\mathcal G}_{(\alpha)}\) and \({\mathcal G}_(\alpha) \cap {\mathcal G}_p \cap {\mathcal G}_{\alpha (6))}\) are radical classes. Some open questions: Are \({\mathcal G}_{(\alpha)} \cap {\mathcal G}_{(\alpha (6))}\) or \({\mathcal G}_p \cap {\mathcal G}_{(\alpha(6))}\) radical classes? Is \({\mathcal G}_{(\alpha)} \cap {\mathcal G}_p \cap {\mathcal G}_{(\alpha (6))}\) a torsion class?
Reviewer: F.Šik (Brno)

MSC:
06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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References:
[1] A. Bigard, K. Keimel, S. Wolfenstein: Groupes et anneaux réticulés, Lecture Notes in Mathematics 608. Springer Verlag, Berlin, 1977.
[2] G. J. M. H. Buskes: Disjoint sequences and completeness properties. Indag. Math (Proc. Netherl. Acad. Sci. A 88) 47 (1985), 11-19. · Zbl 0566.46004
[3] P. Conrad: Lattice ordered groups. Tulane University, 1970. · Zbl 0258.06011
[4] P. Conrad: \(K\)-radical classes of lattice ordered groups. Algebra, Proc. Conf. Carbondale (1980), Lecture Notes in Mathematics 848, 1981, pp. 186-207. · Zbl 0455.06010
[5] M. Darnel: Closure operations on radicals of lattice ordered groups. Czechoslov. Math. J. 37 (1987), 51-64. · Zbl 0624.06022
[6] L. Fuchs: Partially ordered algebraic systems. Oxford, 1963. · Zbl 0137.02001
[7] W. C. Holland: Varieties of \(\ell \)-groups are torsion classes. Czechoslov. Math. J. 29 (1979), 11-12. · Zbl 0432.06011
[8] J. Jakubík: Radical mappings and radical classes of lattice ordered groups. Symposia Math. 31, Academic Press, New York-London, 1977, pp. 451-477.
[9] J. Jakubík: Projectable kernel of a lattice ordered group. Universal algebra and applications, Banach Center Publ. Vol. 9, 1982, pp. 105-112.
[10] J. Jakubík: Kernels of lattice ordered groups defined by properties of sequences. Časopis pěst. matem. 109 (1984), 290-298. · Zbl 0556.06007
[11] J. Jakubík: On some types of kernels of a convergence \(\ell \)-group. Czechoslov. Math. J. 39 (1989), 239-247. · Zbl 0748.06006
[12] J. Jakubík: Closure operators on the lattice of radical classes of lattice ordered groups. Czechoslov. Math. J. 38 (1988), 71-77. · Zbl 0655.06012
[13] J. Jakubík: On a radical class of lattice ordered groups. Czechoslov. Math. J. 39 (1989), 641-643. · Zbl 0713.06008
[14] W. A. J. Luxemburg, A. C. Zaanen: Riesz spaces, Volume I. Amsterdam-London, 1971. · Zbl 0231.46014
[15] J. Martinez: Torsion theory for lattice groups. Czechoslov. Math. J. 25 (1975), 284-292. · Zbl 0321.06020
[16] N. Ja. Medvedev: On the lattice of radicals of a finitely generated \(\ell \)-group. Math. Slovaca 33 (1983), 185-188. · Zbl 0513.06009
[17] R. Sikorski: Boolean algebras, Second edition. Berlin, 1964. · Zbl 0123.01303
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