  As a common non-trivial 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.
