2017
Том 69
№ 9

All Issues

Protasov I. V.

Articles: 23
Brief Communications (Ukrainian)

Descriptive complexity of the sizes of subsets of groups

Banakh T. O., Protasov I. V., Protasova K. D.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1280-1283

We study the Borel complexity of some basic families of subsets of a countable group (large, small, thin, rarefied, etc.) determined by the sizes of their elements. The obtained results are applied to the Czech – Stone compactification $\beta G$ of the group $G$. In particular, it is shown that the closure of the minimal ideal $\beta G$ has the $F_{\sigma \delta}$ type.

Article (English)

Ultrafilters on balleans

Protasov I. V., Slobodianiuk S. V.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1698-1706

A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in terms of companions.

Article (English)

Scattered Subsets of Groups

Banakh T. O., Protasov I. V., Slobodianiuk S. V.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 304-312

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets.

Article (English)

Thin Subsets of Groups

Protasov I. V., Slobodianiuk S. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1245–1253

For a group G and a natural number m; a subset A of G is called m-thin if, for each finite subset F of G; there exists a finite subset K of G such that |F g A| ≤ m for all gG \ K: We show that each m-thin subset of an Abelian group G of cardinality ℵ n ; n = 0, 1,… can be split into ≤ m n+1 1-thin subsets. On the other hand, we construct a group G of cardinality ℵ ω and select a 2-thin subset of G which cannot be split into finitely many 1-thin subsets.

Article (English)

Balleans and G -spaces

Petrenko O. V., Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 344-350

We show that every ballean (equivalently, coarse structure) on a set $X$ can be determined by some group $G$ of permutations of $X$ and some group ideal $\mathcal{I}$ on $G$. We refine this characterization for some basic classes of balleans: metrizable, cellular, graph, locally finite, and uniformly locally finite. Then we show that a free ultrafilter $\mathcal{U}$ on $\omega$ is a $T$-point with respect to the class of all metrizable locally finite balleans on $\omega$ if and only if $\mathcal{U}$ is a $Q$-point. The paper is concluded with а list of open questions.

Article (English)

Relatively thin and sparse subsets of groups

Lutsenko Ie., Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 216-225

Let $G$ be a group with identity $e$ and let $\mathcal{I}$ be a left-invariant ideal in the Boolean algebra $\mathcal{P}_G$ of all subsets of $G$. A subset $A$ of $G$ is called $\mathcal{I}$-thin if $gA \bigcap A \in \mathcal{I}$ for every $g \in G \ \{e\}$. A subset $A$ of $G$ is called $\mathcal{I}$-sparse if, for every infinite subset $S$ of $G$, there exists a finite subset $F \subset S$ such that $\bigcap_{g \in F}gA \in F$. An ideal $\mathcal{I}$ is said to be thin-complete (sparse-complete) if every $\mathcal{I}$-thin ($\mathcal{I}$-sparse) subset of $G$ belongs to $\mathcal{I}$. We define and describe the thin-completion and the sparse-completion of an ideal in $\mathcal{P}_G$.

Article (Russian)

On the Suslin Number of Totally-Bounded Left-Topological Groups

Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1581-1573

For every infinite cardinal α, we construct a zero-dimensional totally-bounded left-topological group with Suslin number α.

Article (English)

Morphisms of Ball's Structures of Groups and Graphs

Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 847-855

We introduce and study two kinds of morphisms between ball's structures related to groups and graphs.

Article (Russian)

Ultrafilters and Decompositions of Abelian Groups

Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 85-93

We prove that every PS-ultrafilter on a group without second-order elements is a Ramsey ultrafilter. For an arbitrary PS-ultrafilter ϕ on a countable group G, we construct a mapping f: G → ω such that f(ϕ) is a P-point in the space ω*. We determine a new class of subselective ultrafilters, which is considerably wider than the class of PS-ultrafilters.

Article (Ukrainian)

Decompositions of direct products of groups

Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1385–1395

We propose a new method for the decomposition of direct products of groups into U subsets. By using this method, we prove the following generalization of the Comfort-van Mill theorem: An arbitrary nondiscrete topological Abelian group with a finite number of second-order elements can be decomposed into a countable number of dense subsets.

Article (Russian)

Ideals and free Pairs in the semigroup β ℤ

Protasov I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 573–580

We prove that the equations ξ+x=mξ+y, x+ξ=y+mξ have no solutions in the semigroup β ℤ for every free ultrafilter ξ and every integer m∈0, 1. We study semigroups generated by the ultrafilters ξ, mξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ℤ.

Article (Ukrainian)

Bourbaki spaces of topological groups

Charyev A., Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 542–549

Article (Ukrainian)

Ochan topologies on the space of closed subgroups

Protasov I. V., Stukotilov V. S.

Full text (.pdf)

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1337–1342

Article (Ukrainian)

Souslin number of the space of subgroups of a locally compact group

Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 654-658

Article (Ukrainian)

Compact subspaces in the space of subgroups of a topological group

Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 600–605

Article (Ukrainian)

Topological groups with a σ-compact space of subgroups

Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 93 – 98

Article (Ukrainian)

The Chabauty topology on the lattice of closed subgroups

Protasov I. V., Tsybenko Yu. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 207 - 213

Article (Ukrainian)

Connectedness of the space of subgroups

Protasov I. V., Tsybenko Yu. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1983. - 35, № 3. - pp. 382 — 385

Article (Ukrainian)

Topological properties of the lattice of subgroups

Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 355 - 360

Article (Ukrainian)

Dualisms of topological Abelian groups

Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1979. - 31, № 2. - pp. 207–211

Article (Ukrainian)

Projections of abelian topological groups

Mukhin Yu. N., Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1978. - 30, № 4. - pp. 551–556

Article (Ukrainian)

Topological dualities of locally compact Abelian groups

Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1977. - 29, № 5. - pp. 625–631

Article (Ukrainian)

Groups with uniquely generated invariant subgroups

Charin V. S., Protasov I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1977. - 29, № 2. - pp. 275–280