Pastukhova I. Ditopological Inverse Semigroups

The dissertation focuses on defining and studying a new class of topological inverse semigroups, called ditopological semigroups. Ditopological inverse semigroups form a natural class of topological inverse semigroups, containing all topological groups, topological semilattices, all uniformizable and compact inverse semigroups. Moreover, this class in preserved by taking inverse subsemigroups and operations of Tychonoff, semidirect and reduced products. In addition, an example of a topological inverse semigroup which is not ditopological is constructed in the dissertation. The main problem of the dissertation was to extend to a non-compact case the embedding theorems obtained by O. Hryniv, who constructed a topological embedding of some Clifford compact inverse semigroups into the products of zero-extensions of topological groups or cones over topological groups. In the dissertation these results are extended to non-compact ditopological Clifford inverse semigroups. In particular, an embedding of a ditopological Clifford inverse U_0-semigroup into the product of maximal semilattice and zero-extensions of maximal groups is constructed. Another result builds an embedding of a ditopological Clifford inverse U-semigroup into the product of maximal semilattice and cones over maximal groups. These embedding theorems have several important implications. One of them provides a criterion of embeddability of a Clifford topological inverse U-semigroup into a compact Clifford inverse semigroup. Besides, the embeddability theorems for ditopological Clifford semigroups are applied to obtain a criterion of metrizability of ditopological Clifford inverse semigroups (by a subinvariant metric) in terms of metrizability of its maximal semilattices and its maximal subgroups. The structure results for Clifford ditopological inverse semigroups allow us to obtain interesting results on the automatic continuity of homomorphisms between ditopological Clifford inverse semigroups. It is reasonable to consider the following automatic continuity problem for Clifford inverse topological semigroups: is the homomorphism between Clifford topological inverse semigroups continuous provided its restrictions to each subsemilattice and each subgroup are continuous? This problem was solved affirmatively by Bowman for compact Clifford topological inverse semigroups with Lawson maximal semilattice and by Yeager for compact Clifford topological inverse semigroups. In the dissertation analogs of the above automatic continuity theorems for non-compact ditopological inverse semigroups are obtained.