Continuity of the inverse in pseudocompact paratopological groups

By a celebrated theorem of Numakura, a Hausdorff compact topological semigroup with two-sided cancellation is a group which has inverse continuous, i.e., it is a topological group. We improve Numakura's Theorem in the realm of non-Hausdorff topological semigroups. This improvement together with some properties of pseudocompact nature in the field of bitopological spaces is used in order to prove that a T-o paratopological group (G, tau) is a (Hausdorff) pseudocompact topological group if and only if (G, tau boolean OR tau(-1)) is pseudocompact or, equivalently, G is G(delta)-dense in the Stone-Cech bicompactification (beta(2)G, (tau) over cap, (tau(-1)) over cap) of (G, tau, tau(-1)). We also present a version for paratopological groups of the renowned Comfort-Ross Theorem stating that a topological group is pseudocompact if and only if its Stone-Cech compactification is a topological group.