Topological Brandt lambda-extensions of topological semigroups

If a topological-algebraic object which is included in another object, is contained in it as a closed subspace, then it is called H-closed. Striving to solve the problem of finding the criterion of the H-closedness and the absolute H-closedness in the category of topological semigroups, a topologically-algebraic extension (namely, the topological Brandt lambda-extension of topological semigroups) was constructed which preserves both H-closedness and absolute H-closedness in the class of topological inverse semigroups. For every infinite cardinal $lambda$, semigroup topologies on Brandt lambda-extensions which preserve H-closedness and absolute H-closedness were constructed. An example of a non H-closed topological inverse semigroup S in the class of topological inverse semigroups such that for any cardinal $\lambda >1$ there exist an absolutely H-closed topological Brandt lambda-extension of the semigroup S in the class of topological semigroups is constructed. Sufficient conditions on topological Brandt extensions for preserving (absolute) H-closedness will be given.

As a consequence of the obtained results, the structure of compact 0-simple topological inverse semigroups, structure of compact and countably compact primitive topological inverse semigroups and structure of pseudocompact completely 0-simple topological inverse semigroups will be described.

Using the construction of topological Brandt lambda-extensions of topological semigroups, an example of the countable absolutely H-closed 0-dimensional metrizable inverse topological semigroup S with an absolutely H-closed ideal I such that the Rees quotient-semigroup S/I is not a topological semigroup will be constructed.