arXiv subjects

Oleksandr Ravsky

Publications and source records attributed to Oleksandr Ravsky.

3 recordsLinked to original sources

arXiv2015

On feebly compact inverse primitive (semi)topological semigroups

We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions when the maximal subgroup of an inverse primitive feebly compact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt $λ^0$-extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse feebly compact topological semigroup is continuous. Also an analogue of Comfort--Ross Theorem is proved for such semigroups: a Tychonoff product of an arbitrary family of primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups is feebly compact. We describe the structure of the Stone-Čech compactification of a Hausdorff primitive inverse countably compact semitopological semigroup $S$ such that every maximal subgroup of $S$ is a topological group.

math.GR↗

arXiv2015

Pseudocompactness, products and topological Brandt $λ^0$-extensions of semitopological monoids

In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, $ω$-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) to\-pological Brandt $λ_i^0$-extensions of semitopological monoids with zero. In particular we show that if $\big\{ \big(B^0_{λ_i}(S_i),τ^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ is a family of Hausdorff pseudocompact to\-pological Brandt $λ_i^0$-extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product $\prod\left\{ S_i \colon i\in\mathscr{I}\right\}$ is a pseudocompact space then the direct product $\prod\big\{ \big(B^0_{λ_i}(S_i),τ^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.

math.GR↗

arXiv2013

On partitions of G-spaces and G-lattices

Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of $X$, we prove that for every partition $X=A_1\cup\dots\cup A_n$ of $X$ into $n\ge 2$ pieces there is a piece $A_i$ of the partition and a finite set $F\subset G$ of cardinality $|F|\le ϕ(n+1):=\max_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$ such that $G=F\cdot Δ(A_i)$ where $Δ(A_i)=\{g\in G:gA_i\cap A_i\notin I\}$ is the difference set of the set $A_i$. Also we investigate the growth of the sequence $ϕ(n)=\max_{1<x<n}\frac{x^{n-x}-1}{x-1}$ and show that $\ln ϕ(n)=nW(ne)-2n+\frac{n}{W(ne)}+\frac{W(ne)}{n}+O\big(\frac{\ln n}n\big)$ where $W(x)$ is the Lambert W-function, defined implicitly as $W(x)e^{W(x)}=x$. This shows that $ϕ(n)$ grows faster that any exponent $a^n$ but slower than the sequence of factorials $n!$.

math.CO↗