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 T 1 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 ∈ I G i is a product of T 1 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.