arXiv subjects
Publications and source records attributed to Igor Protasov.
We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-Čech compactification $βG$ of $G$, we prove, in particular, that the closure of the minimal ideal of $βG$ is of type $F_{σδ}$.
A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean $\mathcal{B}$ on a set $X$, the hyperballean $\mathcal{B}^{\flat}$ is a ballean naturally defined on the set $X^{\flat}$ of all bounded subsets of $X$. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans.
Let $G$, $H$ be groups and $κ$ be a cardinal. A bijection $f:G\to H$ is caled on asymorphism if, for any $X\in[G]^{<κ}$, $Y\in[H]^{<κ}$, there exist $X'\in[G]^{<κ}$, $Y'\in[H]^{<κ}$ such that for all $x\in G$ and $y\in H$, we have $f(Xx)\subseteq Y'f(x)$, $f^{-1}(Yy)\subseteq X'f^{-1}(y)$. For a set $S$, $[S]^{<κ}$ denotes the set $\{S'\subseteq S: |S'|<κ\}$. Let $κ$ and $γ$ be cardinals such that $\aleph_0<κ\leγ$. We prove that any two Abelian groups of cardinality $γ$ are $κ$-asymorphic, but the free group of rank $γ$ is not $κ$-asymorphic to an Abelian group provided that either $κ<γ$ or $κ=γ$ and $κ$ is a singular cardinal. It is known [7] that if $γ= κ$ and $κ$ is regular then any two groups of cardinality $κ$ are $κ$-asymorphic.
Let $G $ be a group of cardinality $κ>\aleph_0 $ endowed with a topology $τ$ such that $|U|=κ$ for every non-empty $U\inτ$ and $τ$ has a base of cardinality $κ$. We prove that $G$ could be factorized $G=AB$ (i.e. each $g\in G$ has unique representation $g=ab$, $a\in A$, $b\in B$) into dense subsets $A,B$, $|A|=|B|=κ$. We do not know if this statement holds for $κ= \aleph_0$ even if $G$ is a topological group.
Every countable topological group $G$ has a closed discrete subset $A$ such that $G=AA^{-1}.$
We say that a topological group $G$ is partially box $κ$-resolvable if there exist a dense subset $B$ of $G$ and a subset $A $ of $G$, $|A|=κ$ such that the subsets $\{ aB: a\in A\}$ are pairwise disjoint. If $G=AB$ then $G$ is called box $κ$-resolvable. We prove two theorems. If a topological group $G$ contains an injective convergent sequence then $G$ is box $ω$-resolvable. Every infinite totally bounded topological group $G$ is partially box $n$-resolvable for each natural number $n$, and $G$ is box $κ$-resolvable for each infinite cardinal $κ, κ<|G|$.
For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $cof(X)=cof(Y)$ and $cov^\sharp(X)=cov^\sharp(Y)$ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of the cardinals $cof(X)$ and $cov^\sharp(X)$. For every limit ordinal $γ$ we shall define a ballean $2^{<γ}$ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality $cf(γ)$ plays a role analogous to the role of the Cantor cube $2^κ$ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to $2^{<γ}$. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube $2^ω$.
Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in \mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset $F$ of $S$ such that $S=F ^{-1} \{x \in S: x A \bigcap A\neq\emptyset\}$ and $|F |\leq n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.
For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-Čech compactifications $βG$ and $βX$ with the sets of all ultrafilters on $G$ and $X$, and apply the natural action of $βG$ on $βX$ to characterize large, thick, thin, sparse and scattered subsets of $X$. We use $G$-invariant partitions and colorings to define $G$-selective and $G$-Ramsey ultrafilters on $X$. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on $ω$, the $T$-points, and study interrelations between these ultrafilters and some classical ultrafilters on $ω$.
Given a semigroup $S$, we introduce relative (with respect to a filter $τ$ on $S$) versions of large, thick and prethick subsets of $S$, give the ultrafilter characterizations of these subsets and explain how large could be some cell in a finite partition of a subset $A\inτ$.
Let $G$ be a group and let $X$ be a transitive $G$-space. We classify the subsets of $X$ with respect to a translation invariant ideal $\mathcal{J}$ in the Boolean algebra of all subsets of $X$, introduce and apply the relative combinatorical derivations of subsets of $X$. Using the standard action of $G$ on the Stone-$\check{C}$ech compactification $βX$ of the discrete space $X$, we characterize the points $p\inβX$ isolated in $Gp$ and describe a size of a subset of $X$ in terms of its ultracompanions in $βX$. We introduce and characterize scattered and sparse subsets of $X$ from different points of view.
We classify the subsets of a group by their sizes, formalize the basic methods of partitions and apply them to partition a group to subsets of prescribed sizes.
We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\inω}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\inω$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.
Let G be a group and let k be a cardinal. A subset A of G is called left (right) k-large if there exists a subset F of G such that |F| < { and G = FA (G = AF). We say that A is k-large if A is left and right k-large. It is known that every infinite group G can be partitioned into countably many \aleph_0-large subsets. On the other hand, every amenable (in particular Abelian) group G cannot be partitioned into > \aleph_0 \aleph_0-large subsets. We prove that every infinite group G of cardinality k can be partitioned into k left- \aleph_1-large subsets and every free group F_k in the infinite alphabet k can be partitioned into k 4-large subsets.
Every infinite group $G$ of regular cardinality can be partitioned $G=A_1\cup A_2$ so that $G\neq FA_1$, $G\neq FA_2$ for every subset $F\subset G$ of cardinality $|F|<|G|$. The first author asked whether the same is true for each group $G$ of singular cardinality. We show that an answer depends on the algebraic structure of $G$. In particular, this is so for each free group but the statement does not hold for every Abelian group $G$ of singular cardinality. As an application, we prove that every Abelian group of singular cardinality k admits maximal translation invariant k-bounded topology that impossible for all groups of regular cardinality.
In 1995 in Kourovka notebook the second author asked the following problem: it is true that for each partition $G=A_1\cup\dots\cup A_n$ of a group $G$ there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some set $F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups.
Given a set $X$ and a family $G$ of self-maps of $X$, we study the problem of the existence of a non-discrete Hausdorff topology on $X$ with respect to which all functions $f\in G$ are continuous. A topology on $X$ with this property is called a $G$-topology. The answer is given in terms of the Zariski $G$-topology $ζ_G$ on $X$, that is, the topology generated by the subbase consisting of the sets $\{x\in X:f(x)\ne g(x)\}$ and $\{x\in X:f(x)\ne c\}$, where $f,g\in G$ and $c\in X$. We prove that, for a countable monoid $G\subset X^X$, $X$ admits a non-discrete Hausdorff $G$-topology if and only if the Zariski $G$-topology $ζ_G$ is non-discrete; moreover, in this case, $X$ admits $2^{\mathfrak c}$ hereditarily normal $G$-topologies.
We prove that for every number k each countable infinite group $G$ admits a partition $G=A\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\subset G$ the sets $KA$ and $KB$ are not thick. The proof is based on the fact that $G$ possesses a syndetic submeasure, i.e., a left-invariant submeasure $μ:\mathcal P(G)\to[0,1]$ such that for each $ε> 1/|G|$ and subset $A\subset G$ with $μ(A)<1$ there is a set $B\subset G\setminus A$ such that $μ(B)<ε$ and $FB=G$ for some finite subset $F\subset G$.