Pyekhtyeryev V. Cross-sections of transformation semigroups.

Thesis is devoted to cross-sections of transformation semigroups. А complete description of all retracts for the full finite transformation semigroup is obtained. For the full transformation semigroup all H- and R-cross-sections are classified. We determine which R-cross-sections are isomorphic. A description of all retracts for the finite inverse symmetric semigroup is given, and also all cross-sections of the infinite inverse symmetric semigroup are described. A description of all R- and L-cross-sections for the full inverse symmetric semigroup is found. An isomorphism criterion for R- (L-) cross-sections of this semigroup is obtained. The second part of the work is devoted to the semigroup of all partial continuous injections of a Hausdorff space. It is shown that this semigroup contains H-, R- or L-cross-sections if and only if the space X is discrete. Thus a criterion of discreteness of a Hausdorff space in terms of transformation semigroups is obtained.