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Locally finite conditions on lattice-ordered groups. (English) Zbl 0688.06011
Suppose \(\alpha\) stands for a certain property or class of \(\ell\)-groups. We say that an \(\ell\)-group G is locally \(\alpha\) if every finitely \(\ell\)-generated \(\ell\)-subgroup of G satisfies \(\alpha\). \({\mathcal S}\) denotes the class of \(\ell\)-groups which are \(\ell\)-isomorphic to an \(\ell\)-group of real-valued step functions, i.e. of functions \(f\in {\mathbb{R}}^ I\) with f(I) finite. The authors study the following seven conditions: (1) Locally G satisfies the ACC on all subgroups. (2) Locally G satisfies the ACC on all \(\ell\)-subgroups. (3) G is locally finitely generated. (4) Locally G has a finite root system of p prime subgroups. (5) Locally G has a finite basis. (6) G is locally finite-valued. (7) For each \(0<x\in G\), \(G(x)/N_ x\in {\mathcal S}\). It is shown that for a nilpotent \(\ell\)-group G the conditions (1) through (7) are equivalent. Moreover some examples are given which indicate how some of the above implications can fail in general (e.g. (4) non \(\Rightarrow\) (1), (5) non \(\Rightarrow\) (4), (6) non \(\Rightarrow\) (5), (7) non \(\Rightarrow\) (6), (6) non \(\Rightarrow\) (3)). Also, they call attention to the \(\ell\)- groups which are characterized by condition (7). Other problems are discussed. Finally, a list of questions is presented.
Reviewer: F.Šik

MSC:
06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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