zbMATH — the first resource for mathematics

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)

06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
Full Text: EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.