Lysetska O. Compact and compact-like semilattices, semigroups and their extensions

The thesis is devoted to the study of the Hausdorff shift-continuous feebly compact topologies exp_n(lambda), the algebraic and topological properties of extensions of monoids by symmetric inverse semigroups of bounded finite rank I_lambda^n(S), the feebly compact topologies on the bicyclic semigroup extension B_omega_F1 in the case when the family F1 consists of the empty set and all singleton subsets of omega. In the thesis all countably compact shift-continuous T1-topologies on the semilattice exp_n(lambda) are described. Also, it is proved that these topologies are semilattice compact topologies for arbitrary integer n>1 and any infinite cardinal. It is constructed a non-semiregular Hausdorff countably pracompact (therefore feebly compact) non-compact shift-continuous topology on the semilattice exp_2(lambda) and it is proved that the semiregular feebly compact semitopological semilattice exp_n(lambda) is compact topological semilattice. It is proved that for any shift-continuous T1-topology tau on exp_n(lambda) the following conditions are equivalent: i) tau is sequentially pracompact; ii) tau is D(omega)-compact. In the thesis construction and algebraic properties of the semigroup extension I_lambda^n(S) of a monoid S up to the modulo of the semigroup S are described. Also, it is introduced the conception of a semigroup with strongly tight ideal series, and conditions of the semigroup I_lambda^n(S) to be a semigroup with (strongly) tight ideal series up to the modulo of the monoid S are found. It is proved that for each compact Hausdorff semitopological monoid S there exists its unique compact topological extension I_lambda^n(S) in the class of the Hausdorff semitopological semigroups and it is described its topology. Also, there are represented the definition of the semigroup B_omega^F1 in the case when the family F1 consists of the empty set and all singleton subsets of omega and investigated its algebraic properties. It is proved that each D(omega)-compact shift-continuous T1-topology on B_omega_F1 is compact and sequentially compact, and moreover it is the Alexandrov one-point compactification of the countable discrete space.