Countably compact paratopological groups

By means of the theory of bispaces we show that a countably compact To paratopological group (G, tau) is a topological group if and only if (G, tau V tau(-1)) is omega-bounded (here tau(-1) is the conjugate topology of tau). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every omega-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. in particular, omega-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.Universidade Federal de São Paulo, Dept Matemat, Inst Matemat & Estad, BR-05311970 São Paulo, BrazilUniv Jaume 1, Dept Matemat, Castellon de La Plana, SpainUniversidade Federal de São Paulo, Dept Matemat, Inst Matemat & Estad, BR-05311970 São Paulo, BrazilWeb of Scienc