Respondent

Pastukhova Iryna Stepanivna

Theme

Ditopological inverse semigroups

Defence Date

02.05.2019

Annotation

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 dissertati-
on these results are extended to non-compact ditopological Clifford inverse semi-
groups. 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 provi-
des a criterion of embeddability of a Clifford topological inverse U-semigroup into
a compact Clifford inverse semigroup. Besides, the embeddability theorems for di-
topological 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 followi-
ng automatic continuity problem for Clifford inverse topological semigroups: is the
homomorphism between Clifford topological inverse semigroups continuous provi-
ded 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.
Keywords: topological inverse semigroup, Clifford topological inverse semi-
group, topological semilattice, topological group, ditopological semigroup.

Dissertation File

Autosummary File