**
The bicyclic semigroup and the semigroup of matrix units as topological
semigroups
**

The properties of the bicyclic semigroup as a topological semigroup were investigated by Wallace, Anderson, Hunter, Koch, Eberhart, Selden, Bertman, and West. The infinite semigroup of matrix units is, in some algebraic sense, an .orthogonal analog. of the bicyclic semigroup. That is why the question naturally arose whether the infinite semigroup of matrix units has, as a topological semigroup, properties similar to those of the bicyclic semigroup. So, the task was to investigate semigroup Hausdorff topologizations of infinite semigroups of matrix units, especially semigroup minimal Hausdorff topologizations, to study its closure in different classes of topological semigroups as well as to research topological embeddings of the infinite semigroup of matrix units into compact topological semigroups, and other compact-like classes of topological semigroups.

Next results will be presented:

* any nonzero element of the infinite semigroup of matrix units is an
isolated point;

* the infinite semigroup of matrix units is algebraically h-closed in the
class of topological inverse semigroup;

* infnite semigroup of matrix units does not embed neigher into a compact
(countably compact) topological semigroup, nor even into Tychonoff
topological semigroup with the pseudo-compact square;

* topological inverse semigroup of matrix units is H-closed if and only if
its band of idempotents is compact;

* and some minimal and minimal inverse semigroup topologies on the infinite
semigroup of matrix units will be described.

As a consequence of the obtained results, the structure of compact 0-simple topological inverse semigroups and the Bohr compactification of the infinite semigroup of matrix units will be described. Application of the results to the investigation of the structure of topological symmetric inverse semigroups with finite domain will be reported. Recent progress in the area will be recounted and some open problems will be posed.