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Igor Protasov

Publications and source records attributed to Igor Protasov.

At least 37 records · Page 2Linked to original sources
arXiv2019

Extremal balleans

A ballean (or coarse space) is a set endowed with a coarse structure. A ballean $X$ is called normal if any two asymptotically disjoint subsets of $X$ are asymptotically separated. We say that a ballean $X$ is ultranormal (extremely normal) if any two unbounded subsets of $X$ are not asymptotically disjoint (every unbounded subset of $X$ is large). Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold. A normal ballean is ultranormal if and only if the Higson$^{\prime}$s corona of $X$ is a singleton. A discrete ballean $X$ is ultranormal if and only if $X$ is maximal. We construct a series of concrete balleans with extremal properties.

math.GN↗
arXiv2019

Functional boundedness of balleans

We pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Also we prove that the Bergman property of groups is a coarse invariant. A special attention is payed to balleans on groups.

math.GR↗
arXiv2019

Constructing balleans

A ballean is a set endowed with a coarse structure. We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets and combs. We analyze the smallest and the largest coarse structures on a set $X$ compatible with a given bornology on $X$.

math.GN↗
arXiv2019

A note on free vector balleans

A vector balleans is a vector space over $\mathbb{R}$ endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean $(X, \mathcal{E})$, there exists the unique free vector ballean $\mathbb{V}(X, \mathcal{E})$ and describe the coarse structure of $\mathbb{V}(X, \mathcal{E})$. It is shown that normality of $\mathbb{V}(X, \mathcal{E})$ is equivalent to metrizability of $(X, \mathcal{E})$.

math.GN↗
arXiv2018

The normality and bounded growth of balleans

By a ballean we understand a set $X$ endowed with a family of entourages which is a base of some coarse structure on $X$. Given two unbounded ballean $X,Y$ with normal product $X\times Y$, we prove that the balleans $X,Y$ have bounded growth and the bornology of $X\times Y$ has a linearly ordered base. A ballean $(X,\mathcal E_X)$ is defined to have bounded growth if there exists a function $G$ assigning to each point $x\in X$ a bounded subset $G[x]\subset X$ so that for any bounded set $B\subset X$ the union $\bigcup_{x\in B}G[x]$ is bounded and for any entourage $E\in\mathcal E_X$ there exists a bounded set $B\subset X$ such that $E[x]\subset G[x]$ for all $x\in X\setminus B$. We prove that the product $X\times Y$ of two balleans has bounded growth if and only if $X$ and $Y$ have bounded growth and the bornology of the product $X\times Y$ has a linearly ordered base. Also we prove that a ballean $X$ has bounded growth (and the bornology of $X$ has a linearly ordered base) if its symmetric square $[X]^{\le 2}$ is normal (and the ballean $X$ is not ultranormal). A ballean $X$ has bounded growth and its bornology has a linearly ordered base if for some $n\ge 3$ and some subgroup $G\subset S_n$ the $G$-symmetric $n$-th power $[X]^n_G$ of $X$ is normal. On the other hand, we prove that for any ultranormal discrete ballean $X$ and every $n\ge 2$ the power $X^n$ is not normal but the hypersymmetric power $[X]^{\le n}$ of $X$ is normal. Also we prove that the finitary ballean of a group is normal if and only if it has bounded growth if and only if the group is countable.

math.GN↗
arXiv2018

A note on bornologies

A bornology on a set $X$ is a family $\mathcal{B}$ of subsets of $X$ closed under taking subsets, finite unions and such that $\cup \mathcal{B}=X$. We prove that, for a bornology $\mathcal{B}$ on $X$, the following statements are equivalent: (1) there exists a vector topology $τ$ on the vector space $\mathbb{V} (X) $ over $\mathbb{R}$ such that $\mathcal{B}$ is the family of all subsets of $X$ bounded in $τ$; (2) there exists a uniformity $\mathcal{U}$ on $X$ such that $\mathcal{B}$ is the family of all subsets of $X$ totally bounded in $\mathcal{U}$; (3) for every $Y \subseteq X$, $Y \notin \mathcal{B}$, there exists a metric $d$ on $X$ such that $\mathcal{B}\subseteq \mathcal{B}_d$, $Y\notin \mathcal{B}_d$, where $\mathcal{B}_d$ is the family of all closed discrete subsets of $(X, d)$; (4) for every $Y \subseteq X$, $Y \notin \mathcal{B}$, there exists $Z\subseteq Y$ such that $Z^{\prime} \notin \mathcal{B}$ for each infinite subset $Z^{\prime}$ of $Z$. A bornology $\mathcal{B}$ satisfying $(4)$ is called antitall. We give topological and functional characterizations of antitall bornologies.

math.GN↗
arXiv2018

Varieties of coarse spaces

A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety $\mathfrak{M}$ contains an unbounded metric space then $\mathfrak{M}$ is the variety of all coarse spaces.

math.GN↗
arXiv2018

Lattices of coarse structures

We consider the lattice of coarse structures on a set $X$ and study metrizable, locally finite and cellular coarse structures on $X$ from the lattice point of view.

math.GN↗
arXiv2018

Free coarse groups

A coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let $(X, \mathcal{E})$ be a coarse space and let $\mathfrak{M}$ be a variety of groups different from the variety of singletons. We prove that there is a coarse group $F_{\mathfrak{M}} (X, \mathcal{E})\in \mathfrak{M}$ such that $(X, \mathcal{E}) $ is a subspace of $F_{\mathfrak{M}} (X, \mathcal{E})$, $X$ generates $F_{\mathfrak{M}} (X, \mathcal{E})$ and every coarse mapping $(X, \mathcal{E}) \longrightarrow (G, \mathcal{E}^{\prime}) $ where $G\in\mathfrak{M}$, $(G, \mathcal{E}^{\prime}) $ is a coarse group, can be extended to coarse homomorphism $F_{\mathfrak{M}} (X, \mathcal{E})\longrightarrow (G, \mathcal{E}^{\prime}) $. If $\mathfrak{M}$ is the variety of all groups, the groups $F_{\mathfrak{M}} (X, \mathcal{E})$ are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.

math.GN↗
arXiv2018

Some more algebra on ultrafilters in metric spaces

We continue algebraization of the set of ultrafilters on a metric spaces initiated in [6]. In particular, we define and study metric counterparts of prime, strongly prime and right cancellable ultrafilters from the Stone-$\check{C}$ech compactification of a discrete group as a right topological semigroup [3]. Our approach is based on the concept of parallelity introduced in the context of balleans in [4].

math.GN↗
arXiv2018

Recent progress in subset combinatorics of groups

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in $G$-spaces, new ideals in the Boolean algebra $\mathcal{P}_{G}$ of all subsets of a group $G$ and in the Stone-$\check{C}$ech compactification $βG$ of $G$ , the combinatorial derivation.

math.CO↗
arXiv2017

Metrically Ramsey ultrafilters

Given a metric space $(X,d)$, we say that a mapping $χ: [X]^{2}\longrightarrow\{0.1\}$ is an isometric coloring if $d(x,y)=d(z,t)$ implies $χ(\{x,y\})=χ(\{z,t\})$. A free ultrafilter $\mathcal{U}$ on an infinite metric space $(X,d)$ is called metrically Ramsey if, for every isometric coloring $χ$ of $[X]^{2}$, there is a member $U\in\mathcal{U}$ such that the set $[U]^{2}$ is $χ$-monochrome. We prove that each infinite ultrametric space $(X,d)$ has a countable subset $Y$ such that each free ultrafilter $\mathcal{U}$ on $X$ satisfying $Y\in\mathcal{U}$ is metrically Ramsey. On the other hand, it is an open question whether every metrically Ramsey ultrafilter on the natural numbers $\mathbb{N}$ with the metric $|x-y|$ is a Ramsey ultrafilter. We prove that every metrically Ramsey ultrafilter $\mathcal{U}$ on $\mathbb{N}$ has a member with no arithmetic progression of length 2, and if $\mathcal{U}$ has a thin member then there is a mapping $f:\mathbb{N}\longrightarrowω$ such that $f(\mathcal{U})$ is a Ramsey ultrafilter.

math.GN↗
arXiv2017

Ideals in $\mathcal{P} _G$ and $βG$

For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $βG$ as a right topological semigroup to introduce and characterize some new ideals in $βG$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $βG$ maximal in $G^*$, $G^* = βG \setminus G$, but this statement does not hold for the group $S_κ$ of all permutations of an infinite cardinal $κ$. We characterize the minimal closed ideal in $βG$ containing all idempotents of $G^*$.

math.GN↗
arXiv2017

Ramsey-product subsets of a group

We say that a subset $S$ of an infinite group $G$ is a Ramsey-product subset if, for any infinite subsets $X$, $Y$ of $G$, there exist $x \in X$ and $y\in Y$ such that $x y \in S$ and $ y x \in S$ . We show that the family $φ$ of all Ramsey-product subsets of $G$ is a filter and $φ$ defines the subsemigroup $ \overline{G^*G^*}$ of the semigroup $G^*$ of all free ultrafilters on $G$.

math.GN↗
arXiv2017

Isometric copies of directed trees in orientations of graphs

For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $ω$-paths of infinite diameter.

math.CO↗
arXiv2017

On recurrence in G-spaces

We introduce and analyze the following general concept of recurrence. Let $G$ be a group and let $X$ be a G-space with the action $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. For a family $\mathfrak{F}$ of subset of $X$ and $A\in \mathfrak{F}$, we denote $Δ_{\mathfrak{F}}(A)=\{g\in G: gB\subseteq A$ for some $B\in \mathfrak{F}, \ B\subseteq A\}$, and say that a subset $R$ of $G$ is $\mathfrak{F}$-recurrent if $R\bigcap Δ_{\mathfrak{F}} (A)\neq\emptyset$ for each $A\in \mathfrak{F}$.

math.GN↗