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Closed convex $$\ell$$-subgroups and radical classes of convergence $$\ell$$-groups. (English) Zbl 0897.06022
A convergence $$\ell$$-group (for short: a cl-group) is any pair $$(G,\alpha)$$ where $$G$$ is an abelian $$\ell$$-group and $$\alpha$$ is a convergence in $$G$$. In the paper under review, the author introduces the notion of a closed convex $$\ell$$-subgroup (a cl-subgroup) of a cl-group and shows that the set $$c(G,\alpha)$$ of all cl-subgroups in $$(G,\alpha)$$ is a Brouwer lattice. Further he defines the degree of a subset of a cl-group and describes the $$\ell$$-subgroups with the degree 0 and 1. Moreover, radical classes of cl-groups are introduced and studied (especially the ordered collection of all radical classes is examined).

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 22C05 Compact groups
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