R-factorizable paratopological groups

For i=1,2,3,3.5, we define the class of Ri-factorizable paratopological groups G by the condition that every continuous real-valued function on G can be factorized through a continuous homomorphism p:G→H onto a second countable paratopological group H satisfying the Ti-separation axiom. We show that the Sorgenfrey line is a Lindelöf paratopological group that fails to be R1-factorizable. However, every Lindelöf totally ω-narrow regular (Hausdorff) paratopological group is R3-factorizable (resp. R2-factorizable). We also prove that a Lindelöf totally ω-narrow regular paratopological group is topologically isomorphic to a closed subgroup of a product of separable metrizable paratopological groups.