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Pedro J. Chocano

Publications and source records attributed to Pedro J. Chocano.

18 recordsLinked to original sources
arXiv2026

On the existence and properties of Alexandroff paratopological groups

We study groups endowed with Alexandroff topologies and show that no non-discrete Alexandroff topology can turn a group into a topological group. This settles negatively the basic existence problem for Alexandroff topological groups. Motivated by this obstruction, we turn to the broader setting of Alexandroff paratopological groups. We establish several fundamental properties of these spaces regarding to connections and compact-like properties among others. As applications, we address two addapted classical open questions within our framework concerning feebly bounded subsets in paratopological groups, proving that connected non-compact Alexandroff paratopological groups offer a positive solution both for products of feebly bounded sets and for the feebly boundedness of $B^2$ when $B$ is a feebly bounded subset.

math.GR↗
arXiv2026

From samples to shape: a finite approach to topological invariants

Given a metric space $Y$ and a compact subspace $X$, we present a method for constructing an inverse sequence of finite topological spaces by sampling points from neighborhoods of $X$ in $Y$. We show that these inverse sequences can be employed to define shape theory and to approximate shape invariants, such as Čech homology groups.

math.GN↗
arXiv2026

Semiflows deforming automorphisms groups

In this short note, we prove that, for any pair of finite groups $G$ and $H$, there exist a finite $T_0$-space $X$ and a semiflow $φ\colon [0,\infty)\times X\to X$ such that $\operatorname{Aut}(X)\cong G$, whereas $\operatorname{Aut}(φ_t(X))\cong H$ for every $t>0$. Thus, the symmetry group of a finite $T_0$-space and those of all its positive-time images can be prescribed independently.

math.GN↗
arXiv2026

On continuous homomorphisms from Alexandroff paratopological groups into topological groups

Alexandroff paratopological groups provide a natural setting in which order-theoretic and algebraic properties interact. In this short note we prove obstruction results when considering continuous homomorphisms from Alexandroff paratopological groups into topological groups. Particularly and, as a consequence of these results, we provide an alternative proof of the fact that the only connected Alexandroff topological group is the one that carries the trivial topology.

math.GN↗
arXiv2025

A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix

Given a finite poset $P$, its zeta matrix $\mathbf Z$ encode fundamental incidence-theoretic information about the order structure. In this paper we introduce and study the \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J - \mathbf Z$, where $\mathbf J$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det(\overline{\mathbf Z}) = (-1)^n \tildeχ(P)$, where $n = |P|$ and $\tildeχ(P)$ is the reduced Euler characteristic of $P$. This provides a new, unexpectedly simple linear-algebraic expression for the Euler characteristic of a poset, complementing existing determinant formulas for matrices derived from incidence relations.

math.CO↗
arXiv2025

Matrix Invariants as Homotopy Invariants in Finite $T_0$-spaces

We establish a bijection between the set of finite topological $T_0$-spaces (or partially ordered sets) and equivalence classes of square matrices. The absolute value of the determinant or the rank of these matrices serve as simple homotopy invariants for the corresponding topological spaces, and consequently, for finite simplicial complexes. To conclude, we explore further relationships and problems concerning finite posets within the context of these matrices.

math.AT↗
arXiv2025

Every group retraction can be realized as a topological retraction

Given a group retraction $r: G \rightarrow H $, we construct a finite topological space $ X_r $ of height 1, together with a topological retraction $\overline{r}: X_r \rightarrow X_r $, such that the group of automorphisms $ \mathrm{Aut}(X_r) $ (or the group of self-homotopy equivalences $ \mathcal{E}(X_r) $) of $X_r$ is isomorphic to $ G $, and $ \mathrm{Aut}(\overline{r}(X_r)) $ (or $\mathcal{E}(\overline{r}(X_r)) $) is isomorphic to $ H$. Moreover, there is a natural map $\overline{r}' : \mathrm{Aut}(X_r) \rightarrow \mathrm{Aut}(\overline{r}(X_r)) $ that coincides with the original group retraction $ r $. As a direct consequence of this construction, we show that height 1 is the minimal height required to realize any finite group as the group of automorphisms (or the group of self-homotopy equivalences) of a finite topological space, except in the case where $ G $ is a symmetric group. In that unique case, the group can be realized by a finite topological space of height 0.

math.AT↗
arXiv2025

Riordan pattern's quest within simplicial complexes

The aim of this paper is twofold. First, we demonstrate how Riordan matrices can be employed to connect well-known concepts in geometric combinatorics, such as $f$-vectors, $h$-vectors $γ$-vectors, in a similar fashion to the McMullen Correspondence, and the Dehn-Sommerville equations, among others. Second, we investigate the combinatorial properties of the topological join operation, both for simplicial complexes and for Alexandroff spaces. Finally, we explore the Riordan matrices arising from the iteration of this topological operation and analyze their properties.

math.CO↗
arXiv2025

Semiflows on finite topological spaces

In this paper, we study flows and semiflows defined on any given finite topological $T_0$-space $X$. We show that there exist non-trivial semiflows on $X$, unless $X$ is a minimal finite space. Specifically, non-trivial semiflows exist if and only if $X$ contains down beat points, and a non-trivial semiflow is essentially a strong deformation retraction. As a consequence of this result, we provide a new and concise proof that the only flow that can be defined on $X$ is the trivial flow. Finally, we discuss the number of different semiflows that can be defined on $X$ in terms of down beat points and other special points.

math.GN↗
arXiv2023

On the dynamics of the combinatorial model of the real line

We study dynamical systems defined on the combinatorial model of the real line. We prove that using single-valued maps there are no periodic points of period 3, which contrasts with the classical and less restrictive setting. Then, we use Vietoris-like multivalued maps to show that there is more flexibility, at least in terms of periods, in this combinatorial framework than in the usual one because we do not have the conditions about the existence of periods given by the Sharkovski Theorem.

math.DS↗
arXiv2022

Characteristic Curves and the exponentiation in the Riordan Lie group: A connection through examples

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Frechet structure of the Riordan group described in [4]. After that we describe some linear dynamical systems in $\mathbb{K}[[x]]$ with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to the Pascal Triangle.

math.DS↗
arXiv2022

A combinatorial description of shape theory

We give a combinatorial description of shape theory using finite topological $T_0$-spaces (finite partially ordered sets). This description may lead to a sort of computational shape theory. Then we introduce the notion of core for inverse sequences of finite spaces and prove some properties.

math.GN↗
arXiv2021

Computational approximations of compact metric spaces

Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a method to approximate the homology groups of $X$ and other algebraic invariants. Finally, we study computational aspects and the implementation of this method.

math.GT↗
arXiv2021

On some topological realizations of groups and homomorphisms

Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely $f$. In addition, realization problems involving homology, homotopy groups and groups of automorphisms are considered.

math.AT↗
arXiv2020

Coincidence theorems for finite topological spaces

We adapt the definition of the Vietoris map to the framework of finite topological spaces and we prove some coincidence theorems. From them, we deduce a Lefschetz fixed point theorem for multivalued maps that improves recent results in the literature. Finally, it is given an application to the approximation of discrete dynamical systems in polyhedra.

math.DS↗