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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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