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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:
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