Factorization properties of paratopological groups

In this article we continue the study of R-factorizability in paratopological groups. It is shown that: (1) all concepts of R-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G is R-factorizable if and only if it is totally ω-narrow and has property ω-QU; (3) every subgroup of a T1 paratopological group G is R-factorizable provided that the topological group G⁎ associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space; (4) if Π=∏i∈IGi is a product of T1 paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π is R-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie.