Popadiuk O. Bicyclic extensions of semigroups and their endomorphims.

The thesis is devoted to the study the algebraic properties of bicyclic extension B_{omega}^{F_n} and the inverse semigroup I_{omega}^{n}(→{conv}) of partial convex order isomorphisms of the linearly ordered set (omega, ≤) of the rank ≤ n, to describe their semigroups of endomorphisms, as well as to study of the existence topologizations the semigroup B_{omega}^{F_n} by compact-like and shift-continuous topologies. The Green relations are described, in particular, it is proved that the Green relations D and J coincide on B_{omega}^{F_n}, the semigroup B_{omega}^{F_n} is isomorphic to the semigroup I_{omega}^{n+1}(→{conv}) of partial convex order isomorphisms of the linearly ordered set (ω, ⩽) of the rank ⩽ n+1, and B_{omega}^{F_n} admits only Rees congruences. The topologization of the semigroup B_{omega}^{F_n} is studied. In particular, it is proved that for any shift-continuous T_1-topology tau on the semigroup B_{omega}^{F_n} every non-zero element of B_{omega}^{F_n} is an isolated point of the topological space (B_{omega}^{F_n},tau). Also it is proved that for any shift-continuous T_1-topology τ on the semigroup B_{omega}^{F_n} the following conditions are equivalent: (1) (B_{omega}^{F_n}, tau) is a compact semitopological semigroup; (2) (B_{omega}^{F_n}, tau) is topologically isomorphic to one-point Alexandroff compactification of the infinite countable discrete space; (3) (B_{omega}^{F_n}, tau) is a compact semitopological semigroup with continuous inversion; (4) (B_{omega}^{F_n}, tau) is an D(ω)-compact space. Injective endomorphisms of the inverse semigroup B_{omega}^{F_n} are described. It is proved that the semigroup of injective endomorphisms of the semigroup B_{omega}^{F_n} is isomorphic to the additive semigroup of non-negative integers (ω, +). The structure of the semigroup End(B_{lambda}) of all endomorphisms of the semigroup λ × λ-matrix units B_{lambda} and it is proved that the semigroup End(B_{lambda}) of all endomorphisms of the semigroup λ × λ-matrix units B_{lambda} is a disjoin union of the semigroup End^{inj}(B_{lambda}) of injective endomorphisms of the semigroup B_{lambda} and the semigroup End^{ann}(B_{lambda}) of all annihilating endomorphisms of the semigroup B_{lambda}. It is proved that the semigroup End(I_{omega}^{n}(→{conv})) of all endomorphisms of the semigroup I_{omega}^{n}(→{conv}) is a disjoint union of the set End^{*}(I_{omega}^{n}(→{conv})) and the ideal End^{1}( I_{omega}^{n}(→{conv})).