Cellular covers of totally ordered abelian groups

Abstract

Some results are similar to those on torsion-free abelian groups (unordered), while others are completely different. For instance, though kernels of o-cellular covers can not be non-zero divisible groups (Lemma 3.1), they may contain non-zero divisible subgroups (Example 3.2); however, the divisible part can not be much larger than the reduced part (Theorem 3.4). There are o-groups, even among the additive subgroups of the rationals, whose o-cellular covers form a proper class (Theorem 4.3).

Torsion homology and cellular approximation

“…After this paper, there has been a lot of interest in classifying cellular covers of abelian groups, as well as in describing the kernels of the covers. In the paper [23], the problem is solved for abelian divisible groups; Fuchs-Göbel [27], relying in part on work of Buckner-Dugas [9], address the reduced case and give an accurate description of which groups can appear as kernels; Farjoun-Göbel-Segev-Shelah [23] have presented groups for which the cardinality of all the possible kernels of cellular covers is unbounded; Fuchs has investigated covers of totally ordered abelian groups [26], and there is also work of Rodríguez and Strüngmann on the cotorsion-free case ( [41], [42]). However, the descriptions of the kernels given in these papers are usually not very explicit, and the authors do not investigate the possible relations of the kernels with the homology of the groups involved.…”

Generators and closed classes of groups

“…This kind of problems has been widely studied in the recent literature, but usually under restrictive hypotheses of finiteness or nilpotency of the groups involved. See for example [5], [19], [20], [24].…”