Quasi-half-factorial subsets of abelian torsion groups

If G is an abelian torsion group with generating subset G0, then by a classical result in the theory of non-unique factorizations, the block monoid B(G0) is a half-factorial monoid if each of its atoms has cross number 1. In this case, G0 is called a half-factorial set. In this note, we introduce the notion of a k-quasi-half-factorial set and show for many abelian torsion groups that G0 k-quasi-half-factorial implies that G0 is half-factorial. We moreover show in general that G0 k-quasi-half-factorial implies that G0 is weakly half factorial, a condition which has been of interest in the recent literature.