arXiv subjects

Adam Bartoš

Publications and source records attributed to Adam Bartoš.

12 recordsLinked to original sources

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 Fra\"iss\'e 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

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 Fra\"iss\'e, 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↗

arXiv2024

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 Fra\"iss\'e theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.

math.LO↗

arXiv2024

Generic Compacta from Relations between Finite Graphs: Theory Building and Examples

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph structure and the relations belong to a given graph category. We relate topological properties of the spectrum to combinatorial properties of the graph categories involved. We then utilise this to exhibit elementary combinatorial constructions of well-known continua as Fra\"iss\'e limits of finite graphs in categories with relational morphisms.

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↗

arXiv2023

Constructing Compacta from Posets

We develop a simple method of constructing topological spaces from countable posets with finite levels, one which applies to all second countable T_1 compacta. This results in a duality amenable to building such spaces from finite building blocks, essentially an abstract analog of classical constructions defining compacta from progressively finer open covers.

math.GN↗

arXiv2022

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 Fra\"iss\'e 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 Fra\"iss\'e theories. We reprove the Fra\"iss\'e-theoretic characterization of the pseudo-arc and we realize every P-adic pseudo-solenoid as a Fra\"iss\'e 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↗

arXiv2021

The weak Ramsey property and extreme amenability

We extend the Kechris--Pestov--Todor\v{c}evi\'c correspondence to weak Fra\"{\i}ss\'{e} 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↗

arXiv2020

Borel complexity up to the equivalence

We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace $\mathcal{K}([0, 1]^ω)$ containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace. We recall that to every analytic subset of $\mathcal{K}([0, 1]^ω)$ there exist an equivalent $G_δ$ subset. Then we show that up to the equivalence open subsets of the hyperspace $\mathcal{K}([0, 1]^ω)$ correspond to countably many classes of metrizable compacta. Finally we use the structure of open subsets up to equivalence to prove that to every $F_σ$ subset of $\mathcal{K}([0, 1]^ω)$ there exists an equivalent closed subset.

math.GN↗

arXiv2018

Tree sums of maximal connected spaces

A topology $τ$ on a set $X$ is called maximal connected if it is connected, but no strictly finer topology $τ^* > τ$ is connected. We consider a construction of so-called tree sums of topological spaces, and we show how this construction preserves maximal connectedness and also related properties of strong connectedness and essential connectedness. We also recall the characterization of finitely generated maximal connected spaces and reformulate it in the language of specialization preorder and graphs, from which it is clear that finitely generated maximal connected spaces are precisely $T_\frac{1}{2}$ tree sums of copies of the Sierpiński space.

math.GN↗

arXiv2018

Constant slope, entropy and horseshoes for a map on a tame graph

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a countably affine tame graph. In particular, we show that in the case of a Markov map $f$ that corresponds to recurrent transition matrix, the condition is satisfied for constant slope $e^{h_{\operatorname{top}}(f)}$, where $h_{\operatorname{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\operatorname{top}}(f)$ is achievable through horseshoes of the map $f$.

math.DS↗

arXiv2018

Lower separation axioms via Borel and Baire algebras

Let $κ$ be an infinite regular cardinal. We define a topological space $X$ to be $T_{κ-Borel}$-space (resp. a $T_{κ-BP}$-space) if for every $x\in X$ the singleton $\{x\}$ belongs to the smallest $κ$-additive algebra of subsets of $X$ that contains all open sets (and all nowhere dense sets) in $X$. Each $T_1$-space is a $T_{κ-Borel}$-space and each $T_{κ-Borel}$-space is a $T_0$-space. On the other hand, $T_{κ-BP}$-spaces need not be $T_0$-spaces. We prove that a topological space $X$ is a $T_{κ-Borel}$-space (resp. a $T_{κ-BP}$-space) if and only if for each point $x\in X$ the singleton $\{x\}$ is the intersection of a closed set and a $G_{<κ}$-set in $X$ (resp. $\{x\}$ is either nowhere dense or a $G_{<κ}$-set in $X$). Also we present simple examples distinguishing the separation axioms $T_{κ-Borel}$ and $T_{κ-BP}$ for various infinite cardinals $κ$, and we relate the axioms to several known notions, which results in a quite regular two-dimensional diagram of lower separation axioms.

math.GN↗