• Наївна спроба отримати алгебраїчну характеризацію напівгруп ендоморфізмів класу F моноунарних алгебр: структура напівгруп ендоморфізмів моноунарних алгебр класу F (початок)
  • The reporter will be discussed on the topics which are annonced in the title of report.

• Моноунарні алгебри: хто вони?
  • Будуть наведені різноманітні приклади моноунарних алгебр з різних галузей математики, зокрема у Проблемі 3n+1, конвеєвскій грі "Житття" та фракталах. По останнім двом прикладам планується КІНО

• The product of a nonempty family of pseudocompact paratopological groups is pseudocompact
  • Will be proved the statement which is announced in the title.

• A specific example of a locally compact cancellative semigroup S which cannot be a subsemigroup of a paratopological group.
  • Answering a next question of I. Guran we construct a locally compact Polish cancellative abelian semigroup S such that all shifts on S are quasi-open (that is, int (a+U) is non-empty for each element a\in S and each nonempty open subset U of S), but S cannot be a subsemigroup of a paratopological group.

• Each discrete subgroup of $S_\omega(X)$ is finite.
  • Let $X$ be an infinite set and $S_\omega(X)$ be the group of all bijections of $X$ with finite supporter, endowed with the topology of the pointwise convergence. Inspired by the talk with I. Guran, we show that each discrete subgroup of $S_\omega(X)$ is finite.

• On monoids of injective partial cofinite selfmaps.
  • We discuss on the semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ of injective partial cofinite selfmaps of infinite cardinal $\lambda$. We show that $\mathscr{I}^{\mathrm{cf}}_\lambda$ is a bisimple inverse semigroup and for every non-empty chain $L$ in $E(\mathscr{I}^{\mathrm{cf}}_\lambda)$ there exists an inverse subsemigroup $S$ of $\mathscr{I}^{\mathrm{cf}}_\lambda$ such that $S$ is isomorphic to the bicyclic semigroup and $L\subseteq E(S)$, we describe the Green relations on $\mathscr{I}^{\mathrm{cf}}_\lambda$ and we prove that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_\lambda$ is a group congruence. We also prove that every Hausdorff locally compact topology $\tau$ on $\mathscr{I}^{\mathrm{cf}}_\lambda$ such that $(\mathscr{I}^{\mathrm{cf}}_\lambda,\tau)$ is a semitopological semigroup, is discrete and we describe the closure of the discrete semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ in a topological semigroup. Finally, we show that the (discrete) semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ cannot embed into a compact-like topological semigroup for any infinite cardinal $\lambda$, and we construct two non-discrete Hausdorff topologies which turn $\mathscr{I}^{\mathrm{cf}}_\lambda$ into a topological inverse semigroup.

• Pseudocompact topological Brandt $\lambda^0$-extensions of semitopological semigroups
  • We introduce pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt $\lambda^0$-extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological semigroups in the class of semitopological semigroups and establish the structure of such extensions.