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).


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