R-factorizable paratopological groups

For i = 1, 2. 3, 3.5, we define the class of R-i-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 T-i-separation axiom. We show that the Sorgenfrey line is a Lindelof paratopological group that fails to be R-1-factorizable. However, every Lindelof totally omega-narrow regular (Hausdorff) paratopological group is R-3-factorizable (resp. R-2-factorizable). We also prove that a Lindelof totally omega-narrow, regular paratopological group is topologically isomorphic to a closed subgroup of a product of separable metrizable paratopological groups. (C) 2009 Elsevier B.V. All rights reserved.