On the continuity of inversion in countably compact inverse topological semigroups

Let S be a countably compact Hausdorff space endowed with a continuous semigroup operation turning S into an inverse semigroup. It is shown that the inversion inv : x bar right arrow x(-1) in S is continuous provided one of the following conditions is satisfied: (1) the space S is sequential, (2) the semigroup S is Clifford, inversely regular, and topologically periodic, (3) the semigroup S is Clifford, topologically periodic and the square S x S is regular and countably compact. These results are close to the best possible since there is an example of a quasi-regular sequentially compact commutative inverse topological semigroup with discontinuous inversion.