On some completeness properties for lattice ordered groups.

*(English)*Zbl 0835.06019From 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)

##### Keywords:

uniform completeness; abelian \(\ell\)-groups; projectable \(\ell\)-groups; lattice ordered groups; radical class; torsion class
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