Semi-generalized co-Bassian groups

As a common nontrivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of $p$-torsion groups and groups of finite torsion-free rank by showing that these groups can be completely determined in terms of generalized finite $p$-ranks and also depends on their quotients modulo the maximal torsion subgroup. Surprisingly, for $p$-primary groups, the concept of a semi-generalized co-Bassian group is closely related to that of a generalized co-Bassian group.