Cardinal invariants and RR-factorizability in paratopological groups

In this paper, cardinal invariants and RR-factorizability in paratopological groups are studied. The main results are that (1) w(G)=ib(G⁎)×χ(G)w(G)=ib(G⁎)×χ(G) holds for every paratopological group G; (2) every paratopological group G   satisfies |G|⩽2ib(G⁎)ψ(G)|G|⩽2ib(G⁎)ψ(G); (3) nw(G)=Nag(G)×ψ(G)nw(G)=Nag(G)×ψ(G) is valid for every completely regular paratopological group G; (4) a completely regular paratopological group G   is R2R2-factorizable (resp. R3R3-factorizable) if and only if it is a totally ω-narrow paratopological group with property ω-QU   and Hs(G)⩽ωHs(G)⩽ω (resp. Ir(G)⩽ωIr(G)⩽ω); (5) if G   is a completely regular R2R2-factorizable (resp. R3R3-factorizable) paratopological group and p:G→Kp:G→K an open homomorphism onto a paratopological group K   such that p−1(e)p−1(e) is countably compact, then K   is R2R2-factorizable (resp. R3R3-factorizable), which gives a partial answer to the question posed by M. Sanchis and M.G. Tkachenko (2010) [17].