arXiv subjects

Wiesław Kubiś

Publications and source records attributed to Wiesław Kubiś.

At least 19 recordsLinked to original sources

arXiv2026

Homogeneity of profinite graphs

Profinite graphs are inverse limits of sequences of nonempty finite graphs. There exists a ``generic'' graph $\Pe$ in this category, although its edge relation is rather poor: only isolated vertices and isolated edges. On the other hand, its topological structure (besides being homeomorphic to the Cantor set) is not trivial, as we show in this note. We prove a Knaster--Reichbach type result: Every topological isomorphism $h\colon K\to L$ where $K$ and $L$ are nowhere dense closed subsets of $\Pe$ consisting of isolated vertices can be extended to a topological automorphism of the whole graph $\Pe$.

math.GN↗

arXiv2026

Universal homogeneous two-sorted ultrametric spaces

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and that the limit is the countable rational Urysohn ultrametric space $\mathbb{U}$. The space $\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\overline{\mathbb{U}}$ is dc-universal for all separable ultrametric spaces, which is in contrast with the situation of classical ultrametric spaces and isometric embeddings, where no such universal space can exist. We study further properties of $\mathbb{U}$, of its variants, and of its automorphism group, which is richer than its group of isometries. In particular, we provide two types of tree representations of the two-sorted ultrametric spaces, discuss connections to valued fields, and characterize the automorphism group of $\mathbb{U}$ as the semidirect product of a group of order preserving bijections and a group of isometries. Furthermore, we show universality of $\operatorname{Aut}(\mathbb{U})$ and identify its universal minimal flow.

math.LO↗

arXiv2026

Generic dc-automorphisms of two-sorted ultrametric spaces

We continue to study ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances, together with the dc-embeddings, which we introduced in our earlier paper "Universal homogeneous two-sorted ultrametric spaces". The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé whose limit we denote by $\mathbb{U}$. The main result of the article is that $\operatorname{Aut}(\mathbb{U})$ has a comeager conjugacy class. For that we show the cofinal amalgamation property of partial automorphisms and characterize amalgamation bases. In fact we develop a general strategy for showing cofinal amalgamation property for a broad class of categories. Furthermore, we show that there is no generic pair of automorphisms, we provide a detailed description of single orbits under dc-automorphisms, and we prove that any finite partial dc-automorphism, even in the presence of other orbits, can be extended to one that is closed or monotone.

math.LO↗

arXiv2026

Categorical approach to graph limits

We define and study a natural category of graph limits. The objects are pairs $(π,μ)$, where $π$ (the distribution of vertices) is an abstract probability measure on some abstract measurable space $(X,\mathcal{A})$ and $μ$ (the distribution of edges) is an abstract finite measure on the square $(X,\mathcal{A})^2$. Morphisms are random maps between the underlying measurable spaces which preserve the distribution of vertices as well as the distribution of edges. We also define a convergence notion (inspired by s-convergence) for sequences of graph limits. We apply tools from category theory to prove the compactness of the space of all graph limits.

math.CO↗

arXiv2026

Hereditarily indecomposable continua as generic mathematical structures

We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set of primes P) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player wins if the limit of this sequence is homeomorphic to a concrete (fixed in advance) space, called generic whenever the second player has a winning strategy. For this purpose, we develop a new robust approximate Fraïssé theory in the context of MU-categories, a generalization of metric-enriched categories, suitable for working directly with continuous maps between metrizable compacta. Our framework extends both the classical and projective Fraïssé theories. We reprove the Fraïssé-theoretic characterization of the pseudo-arc and we realize every P-adic pseudo-solenoid as a Fraïssé limit of a suitable category of continuous surjections on the circle. Moreover, we show that, when playing the game with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic, while the universal pseudo-solenoid is generic over all surjections between circle-like continua. This gives a complete classification of generic continua over full non-trivial subcategories of connected polyhedra with continuous surjections.

math.GN↗

arXiv2025

Uncountable homogeneous structures

We study the existence of uncountable first-order structures that are homogeneous with respect to their finitely generated substructures. In many classical cases this is either well-known or follows from general facts, for example, if the language is finite and relational then ultrapowers provide arbitrarily large such sturctures. On the other hand, there are no general results saying that uncountable homogeneous structures with a given age exist. We examine the monoid of self-embeddings of a fixed countable homogeneous structure and, using abstract Fraïssé theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.

math.LO↗

arXiv2024

The weak Ramsey property and extreme amenability

We extend the Kechris--Pestov--Todorčević correspondence to weak Fra\"ıssé categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders, and categories of strong embeddings of trees.

math.LO↗

arXiv2024

Evolution systems: A framework for studying generic mathematical structures

We introduce the concept of an abstract evolution system, which provides a convenient framework for studying generic mathematical structures and their properties. Roughly speaking, an evolution system is a category endowed with a selected class of morphisms called transitions, and with a selected object called the origin. We illustrate it by a series of examples from several areas of mathematics. We formulate sufficient conditions for the existence of the unique "most complicated" evolution. In case the evolution system "lives" in model theory and nontrivial transitions are one-point extensions, the limit of the most complicated evolution is known under the name Fraisse limit, a unique countable universal homogeneous model determined by a fixed class of finitely generated models satisfying some obvious axioms. Evolution systems can also be viewed as a generalization of abstract rewriting systems, where the partially ordered set is replaced by a category. In our setting, the process of rewriting plays a nontrivial role, whereas in rewriting systems only the result of a rewriting procedure is relevant. An analogue of Newman's Lemma holds in our setting, although the proof is a bit more delicate, nevertheless, still based on Huet's idea using well founded induction.

math.LO↗

arXiv2024

On generic topological embeddings

We show that an embedding of a fixed 0-dimensional compact space $K$ into the Čech--Stone remainder $ω^*$ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from $K$ to compact metric spaces. Using Fraïssé theory we get a few well know theorems about Čech--Stone remainder. We establish the following: -- an ultrametric space $K$ of weight $κ$ can be uniformly embedded into $κ^κ$ as a uniformly nowhere dense subset, -- every uniform homeomorphism of uniformly nowhere dense sets in $κ^κ$ can be extended to a uniform auto-homeomorphism of $κ^κ$, -- every uniformly nowhere dense set in $κ^κ$ is a uniform retract of $κ^κ$. If we assume that $κ$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption.

math.GN↗

arXiv2024

Homogeneous isosceles-free spaces

We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to isometry as vector spaces over the two-element field, endowed with an injective norm. Using isosceles-free decompositions, we provide bounds on the maximal number of distances in arbitrary homogeneous finite metric spaces.

math.LO↗

arXiv2024

Chain union closures

We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. We show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness. Further, we introduce and study the notions of chain union stability and of chain union rank, which measure how often the process of closing a ball space under all unions of chains has to be iterated until a ball space is obtained that is closed under unions of chains.

math.LO↗

arXiv2022

Monotone mappings and lines

We study betweenness preserving mappings (we call them \emph{monotone}) defined on subsets of the plane. Once the domain is a convex set, such a mapping is either the restriction of a homography, or its image is contained in the union of a line and a single point, or its image consists of five points, one of them being between two disjoint pairs of the other four points. We also show that an open planar set cannot be mapped in a one-to-one monotone way into the real line. From this we deduce that a one-to-one monotone mapping from a convex planar set with nonempty interior is necessarily a partial homography. Finally, we prove that a set consisting of three pairwise non-parallel lines does not admit a one-to-one monotone mapping into the real line, while on the other hand a set consisting of three closed line segments intersecting at a single point does admit such a mapping.

math.MG↗

arXiv2022

Evolutions of finite graphs

Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We also study the corresponding profinite graphs. Finally, we present a concrete model of the projectively universal profinite graph, the projective Fraisse limit of finite graphs, showing in particular that it has a dense subset of isolated vertices.

math.CO↗

arXiv2022

Homogeneity of abstract linear spaces

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that every finite linear space embeds into a finite projective plane.

math.LO↗

arXiv2021

On a topological Ramsey Theorem

We introduce natural strengthenings of sequential compactness called the $r$-Ramsey property for each natural number $r\geq 1$. We prove that metrizable compact spaces are $r$-Ramsey for all $r$ and give examples of compact spaces that are $r$-Ramsey but not $r+1$-Ramsey for each $r\geq 1$ (assuming CH for all $r>1$

math.GN↗

arXiv2021

Examples of weak amalgamation classes

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

math.LO↗

arXiv2020

Wallman Duality for Semilattice Subbases

We extend Wallman's classic duality from lattice bases to semilattice subbases and from compact to locally closed compact spaces. Moreover, we make this duality functorial via appropriate relational morphisms.

math.GN↗

arXiv2020

Universal AF-algebras

We study the approximately finite-dimensional (AF) $C^*$-algebras that appear as inductive limits of sequences of finite-dimensional $C^*$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra $\mathcal A_\mathfrak{F}$ with the property that any separable AF-algebra is isomorphic to a quotient of $\mathcal A_\mathfrak{F}$. Equivalently, by Elliott's classification of separable AF-algebras, there are surjectively universal countable scaled (or with order-unit) dimension groups. This universality is a consequence of our result stating that $\mathcal A_\mathfrak{F}$ is the Fra\"ıssé limit of the category of all finite-dimensional $C^*$-algebras and left-invertible embeddings. With the help of Fra\"ıssé theory we describe the Bratteli diagram of $\mathcal A_\mathfrak{F}$ and provide conditions characterizing it up to isomorphisms. $\mathcal A_\mathfrak{F}$ belongs to a class of separable AF-algebras which are all Fra\"ıssé limits of suitable categories of finite-dimensional $C^*$-algebras, and resemble $C(2^\mathbb N)$ in many senses. For instance, they have no minimal projections, tensorially absorb $C(2^\mathbb N)$ (i.e. they are $C(2^\mathbb N)$-stable) and satisfy similar homogeneity and universality properties as the Cantor set.

math.OA↗