math.GN — Pith

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math.GN 2026-07-03

Cone domains are RB-domains only when their cone is simplicial

by Yuxu Chen

Cone domains separate FS-domains from RB-domains

D_C is an RB-domain iff C is simplicial, giving FS-domains that are not RB-domains for any non-simplicial proper cone.

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Let $C$ be a closed, convex, pointed and generating cone in a finite-dimensional real vector space $V$, and let $ D_C=(-C)\cup\{\bot\}$ be the negative cone with a new least element, ordered by the cone order. Keimel proved that these cone domains are FS-domains and asked whether they are always retracts of bifinite domains. We give a sharp answer: \[D_C\text{ is an RB-domain}\quad\Longleftrightarrow\quad C\text{ is simplicial}. \] Thus every non-simplicial proper cone gives an FS-domain which is not an RB-domain. The proof converts the RB approximation property into finite-valued $C$-isotone approximations of the identity. The analytic obstruction is elementary and finite-dimensional: first in Euclidean space, cone-upper sets are represented, up to null sets, as Lipschitz epigraphs; Rademacher's theorem, Fubini's theorem and integration by parts then force the matrix tested against any finite-valued isotone map to lie in the cone generated by the positive rank-one operators $v\otimes\ell$, $v\in C$, $\ell\in C^*$. If such maps approximate the identity, the identity operator lies in this rank-one cone, which is possible exactly when the cone is simplicial. This answers Keimel's question in the negative for the Lorentz cone and other non-simplicial cones.

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math.CO 2026-07-03

Poset powerdomain is RB-domain iff Hasse graph is tree with least element

by Yuxu Chen, Hui Kou +1 more

Characterizing finite posets whose probabilistic powerdomain are RB-domains

Classification shows probabilistic powerdomain fails to preserve RB-domains, with diamond as counterexample.

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We classify the finite posets whose probabilistic powerdomain is an RB-domain. For a finite nonempty poset $P$, let $\Vone(P)$ be the probability powerdomain of $P$, which is the probability simplex ordered by the stochastic order. We prove that $\Vone(P)$ is an RB-domain if and only if $P$ has a least element and the undirected Hasse graph of $P$ is a tree. Consequently, the probabilistic powerdomain does not preserve RB-domains; the four-point diamond gives a finite counterexample. The proof separates two obstructions. First, if $P$ has no least element, then the face of probability measures supported on the minimal points must be fixed pointwise by every deflation below the identity. Secondly, once a least element exists, the Hasse graph is connected, and a cycle in it makes the local stochastic cone non-simplicial. A Euclidean finite-step cone argument then rules out the finite-valued monotone approximations supplied by the RB property.

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math.DS 2026-07-03

Homeomorphism groups of pseudo-solenoids have non-metrizable minimal flows

by Jan Boronski, Aleksandra Kwiatkowska

Universal minimal flows of the homeomorphism groups of pseudo-solenoids are non-metrizable

The result covers the pseudo-circle and shows these groups act on spaces that cannot be metrized.

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We note that homeomorphism groups of all pseudo-solenoids, including the pseudo-circle, have non-metrizable universal minimal flows.

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math.GN 2026-07-03

Profiniteness equals residual finiteness for compact residuated lattices

by Jiang Yang, Pengfei He +1 more

Filter-induced linear topologies on residuated lattices: Hausdorffness, profiniteness, and finiteness conditions

The same equivalence holds for closed subdirect products of finite discrete structures under filter-induced linear topologies.

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We study linear topologies on residuated lattices generated by systems of filters, with emphasis on the uniform structures and separation properties that they determine. A down-directed family of filters gives a natural compatible uniformity, and the associated topology makes the residuated lattice into a topological algebra. We characterize Hausdorffness by the triviality of the intersection of the underlying filter system. For compact topological residuated lattices, we prove the equivalence between topological profiniteness, residual finiteness, and representation as a closed subdirect product of finite discrete residuated lattices. We also analyze the descending chain condition ($DCC$) on filters. Under $DCC$, every filter system has a least element; hence every zero-dimensional linear topology is induced by a single filter, and the canonical map from filters to zero-dimensional linear topologies is bijective. This gives a corrected form of earlier representation arguments and identifies precisely where $DCC$ is required. Finally, working throughout in $\mathrm{ZFC}$, we give a sufficient criterion for the existence of non-discrete Hausdorff linear topologies, illustrated by the G\"{o}del algebra.

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math.GN 2026-07-03

Weak ball of nonseparable Hilbert space homeomorphic to its positive part

by Antonio Avilés

Homeomorphism between close relatives of Hilbertian balls

The same ball is also homeomorphic to its product with the Hilbert cube, and all related B(κ,a,b) spaces coincide up to homeomorphism

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We present a solution to some problems posed by the author and Kalenda. We show that the closed ball of nonseparable Hilbert space in its weak topology is homeomorphic to its positive part, as well as to its product with the Hilbert cube. In the separable setting we obtain that there is a weak homeomorphism of the closed unit ball of $\ell_2$ onto its positive part that preserves the norm, and via a result of Dijkstra and van Mill, the same is true for the ball of $\ell_\infty=\ell_1^*$ in the weak$^*$ topology. All spaces $B(\kappa,a,b)$ considered by the author and Kalenda are shown to be homeomorphic. The solution has been found by AI (Chatgpt 5.5), the role of the author has been to ask the right questions, check and understand the answers, and adapt the writing to his personal human taste.

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math.GR 2026-07-02

This paper shows that homeomorphism groups of countable Stone spaces fall into exactly…

by George Domat, Hannah Hoganson +1 more

Coarse geometry of homeomorphism groups: Classifying countable Stone spaces

The three boundedness classes of homeomorphism groups of countable Stone spaces are exactly the coarse equivalence classes, with the middle…

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Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to quasi-isometry. In a previous paper, we placed the homeomorphism groups of countable Stone spaces into three classes: coarsely bounded, unbounded yet generated by a coarsely bounded set, and unbounded but not generated by any coarsely bounded set. Now we show that these are the coarse equivalence classes: Any two groups within one of these classes are in fact coarsely equivalent. Furthermore, we show that groups in the second class are quasi-isometric to the Hamming cube, the space comprising infinite binary sequences with finitely many nonzero entries equipped with the Hamming distance. As part of the proof, we show that infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent.

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math.GN 2026-07-02

Δ-spaces and Q-spaces match measurable cardinal strength

by János Balázs Ivanyos, Ákos Székely

New results about Q and Delta-spaces

Equiconsistency settles consistency questions and bounds size of Lindelöf Q-spaces with weight at most the continuum.

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A topological space $X$ is called a $Q$-space if every subset of $X$ is a $G_\delta$-set, and $X$ is a $\Delta$-space if for any decreasing sequence $\{D_n : n \in\omega\}$ of subsets of $X$ with empty intersection there is a decreasing sequence $\{U_n : n \in \omega\}$ of open sets with empty intersection such that $D_n \subseteq U_n$ for all $n \in\omega$. Our main result shows that the following statements are equiconsistent: (1) There exists a measurable cardinal; (2) There exists a crowded Baire $T_1$ $\Delta$-space; (3) There exists a crowded Baire $T_4$ $Q$-space; (4) There exists a $T_1$ $\Delta$-space admitting a strictly positive probability measure vanishing on points; (5) There exists a $T_3$ $Q$-space admitting a strictly positive probability measure vanishing on points. This provides complete answers to some problems and partial answers to other problems that have recently appeared in the literature. We also prove a new result concerning Lindel\"of $Q$-spaces: if $X$ is a $T_3$ Lindel\"of $Q$-space with $w(X)\leq \mathfrak c$, then $|X|<\operatorname{cf}(\mathfrak c)$. This yields a number of nonexistence results for large Lindel\"of, locally compact, compact, and countably compact $Q$-spaces.

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math.GN 2026-07-02

Planar disk domain is FS but not RB

by Yuxu Chen, Hui Kou +1 more

FS-domains are not always RB-domains

This supplies the first concrete counterexample to the conjecture that FS-domains and RB-domains coincide.

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We prove that Lawson's planar closed-disk domain is not an RB-domain. This domain is the dcpo of all closed disks in the Euclidean plane, together with the whole plane as bottom, ordered by reverse inclusion. Since this domain is an FS-domain, it gives a concrete example of an FS-domain which is not an RB-domain, answering negatively the long-standing open problem in domain theory of whether FS-domains and RB-domains are identical.

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math.LO 2026-07-01

Halo operator equals ω-accumulation points on every space

by Yoàv Montacute

Halo Semantics for Modal Logic

The resulting operator satisfies axiom 4 without separation axioms and yields completeness of K4 for all infinite spaces.

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In nonstandard analysis the halo of a point in a topological space is the intersection of the nonstandard extensions of all its open neighbourhoods. We define a parametric family of modal operators from the halo by varying which elements of the nonstandard extension are admitted as witnesses, and identify four canonical instances. Two recover well-known modalities: the topological closure and the Cantor derivative. A third reduces to Kripke semantics over the specialisation preorder. The fourth, purely nonstandard instance admits only nonstandard witnesses. The Transfer Principle forces it to coincide with the $\omega$-accumulation point operator, a classical topological notion not previously studied in modal logic. Unlike the Cantor derivative, the $\omega$-accumulation operator maps arbitrary sets to closed sets without any separation axiom, yielding an $\omega$-Cantor-Bendixson decomposition on all topological spaces. Axiom 4 holds universally, again without separation conditions. We prove that K4 is the complete logic over infinite spaces, and GL over infinite $\omega$-scattered spaces.

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math.CO 2026-07-01

646099441937791106493755218560442089979 labeled posets on 19 points

by Rafael Ayala

The number of labeled partial orders and topologies on 19 points

The 39-digit total extends the OEIS sequence A001035 and supplies the labeled topology count for 19 points.

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We report the exact value of the number of labeled partially ordered sets (equivalently, labeled $T_0$ topologies) on 19 points, P(19) = 646099441937791106493755218560442089979, a 39-digit integer extending OEIS A001035, whose largest previously computed term was P(18) (Brinkmann and McKay). By the Stirling transform we also obtain the number of labeled topologies on 19 points, A000798(19) = 689054943207246404281592791142107048261. Our route is the Ern\'e-Stege moment reduction, which expresses P(19) through a few sums of antichain counts over the posets on at most 16 points. All of these are available from the posets on at most 15 points (whose number is catalogued, and which standard software generates on demand), except a single moment over the 16-point posets. That moment is obtained not by enumerating the 16-point posets but by inserting a single element into the 15-point ones, with a per-parent kernel that advances the sum at the cost of computing the parent's own antichain count. The result passes several independent checks, among them the residue predicted by the modular periodicity of A001035 and the recovery from the same sweep of the known count P(16) and the Ern\'e-Stege moments G(16,1) and G(16,2). We also report the moments G(16,3) and G(16,4), the latter an input to the analogous computation for 20 points.

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math.GN 2026-06-30

Betti curves prove more stable than persistence images for single-cell leukemia classifica

by Rocío Picón-González, Salvador Chulián +3 more

A Systematic Framework for Evaluating Topological Representations in Single-Cell Classification

A two-level test of group separability and model prediction finds large performance gaps among topological representations in pediatric ALL

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Recent advances in biomedicine generate high-dimensional single-cell data that describe cellular heterogeneity with unprecedented detail, but their geometric complexity and non-linear structure often limit the effectiveness of conventional statistical tools. Topological Data Analysis (TDA) provides a mathematical framework for characterizing the shape of data through persistent homology, which extracts structural features such as connected components and cycles across multiple scales. In this work, we propose a systematic two-level framework for evaluating topological representations in high-dimensional single-cell classification. The first level ($R_1$) performs statistical screening of topological descriptors based on separability between clinical groups, whereas the second level ($R_2$) evaluates their predictive utility in supervised classification models. This design makes it possible to compare representations not only in terms of discriminative performance, but also in terms of robustness to analytical choices. We illustrate the framework using bone marrow flow cytometry data from pediatric acute lymphoblastic leukemia, with a particular focus on relapse stratification. The results show that different topological representations vary substantially in both statistical separability and predictive stability, with Betti Curves and Persistence Silhouettes showing more robust behavior than Persistence Images in this cohort. Overall, the study provides a reproducible methodological framework for the systematic comparison of topological descriptors in complex biomedical point clouds.

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math.GR 2026-06-30

Locally finite groups make every bijective CA reversible

by Jiang Yang

A Reversibility Characterization of Locally Finite Groups by Cellular Automata

A group admits a non-reversible bijective cellular automaton over some alphabet precisely when it fails to be locally finite.

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For cellular automata over finite alphabets, bijectivity already implies reversibility. Over infinite alphabets this implication may fail, and the remaining obstruction in the periodic case was recorded by Ceccherini-Silberstein and Coornaert as Open Problem 2 in \emph{Cellular Automata and Groups}. We prove an exact group-theoretic characterization. A group $G$ is locally finite if and only if, over every alphabet, every bijective cellular automaton $A^G\to A^G$ is reversible. Equivalently, if $G$ is not locally finite, then for every infinite alphabet $A$ there exists a bijective cellular automaton $A^G\to A^G$ whose inverse is not a cellular automaton. The counterexample is already obtained on a countable alphabet. Its local rule has a rank track, a direction track and a binary data track; the forward map is triangular along finite directed chains of arbitrary length, so its inverse is defined pointwise but has no uniform finite memory. As a consequence, Open Problem 2 has an affirmative answer, and the periodicity hypothesis is unnecessary for the negative direction.

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math.CT 2026-06-29

SFC-categories fix sober spaces as adjunction fixed points

by Rui Prezado, Anna Laura Suarez

A general framework for the faithful pointfree representation of T₀-spaces

A general framework for pointfree representations of T0-spaces recovers the Banaschewski-Pultr sober and TD characterizations in a broader s

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We introduce a general framework for studying natural contravariant adjunctions that refine the adjunction between frames and spaces so that the fixpoints are $T_0$-spaces. Our objects of study are \textit{spatializable $\mathbf{Frm}$-concrete categories}, or \textit{SFC-categories}. These consist of a faithful functor $\mathcal O:\mathcal C\to \mathbf{Frm}$ equipped with an object $2_{\mathcal C} \in \mathcal C$, satisfying compatibility conditions that ensure that $(2_{\mathcal C},\mathbb{S})$ forms a dualizing object in the sense of Porst and Tholen, where $\mathbb{S}$ denotes the Sierpi\'nski space. Three important instances of pointfree $T_0$ spaces present in the literature fit into this framework: strictly zero-dimensional biframes, MT-algebras, and Raney extensions. We show SFC-categories are assembled in an ordered category -- a category enriched in preordered sets -- whose morphisms are suitable functors which preserve certain initial liftings. SFC-categories induce natural dual adjunctions, and morphisms between them will respectively induce suitable morphisms between these adjunctions. Motivated by the characterization of sober spaces as maximal objects in the fibers of $\Omega:\mathbf{Top}\to \mathbf{Frm}^{\mathsf{op}}$, and of $T_D$-spaces as the minimal ones, due to Banaschewski and Pultr, we study initial and terminal objects of fibers for an arbitrary SFC-category. We prove that the natural adjunction for fiber-initials has exactly the sober spaces as fixpoints, while for fiber-terminals contains at most $T_D$-spaces, recovering their results of in a much more general setting.

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math.GN 2026-06-29

Remote points of the reals form an ω-bounded space

by Serhii Bardyla, Jaroslav Šupina

New approaches to remote points

They embed into the Stone space of the Boolean algebra generated by open and nowhere dense sets.

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For a given Tychonoff space $X$, a point $p\in \beta(X)\setminus X$ is called {\em remote} if $p$ is not in the closure of any nowhere dense subset of $X$. In this paper, we characterize spaces with remote points in terms of certain topological ultrafilters, measures, and compact-like properties corresponding to the ideal consisting of nowhere dense sets. It is shown that the space of remote points is homeomorphic to a subspace of the Stone space taken over the smallest Boolean algebra containing all open and nowhere dense sets. Also, we show that the space of remote points of $\mathbb R$ is $\omega$-bounded.

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math.GN 2026-06-26

Scattered compact spaces fail the two-disjoint-copies property

by Jerzy Kąkol, Ondřej Kurka +1 more

The two-disjoint-copies property for compact spaces, homogeneity and connection with C_p-theory

ZFC examples show some perfect compacts also lack it, making the property equivalent to uncountability among metric cases.

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A Tychonoff space $X$ has the two-disjoint-copies property (2DCP) if there exists a sequence $(K_n)_{n\in\omega}$ of non-empty compact subsets of $X$ such that each $K_n$ contains two disjoint subsets homeomorphic to $K_{n+1}$. Banakh, K\k{a}kol and \'Sliwa showed that 2DCP yields an infinite-dimensional metrizable quotient of $C_p(X)$, while it is still a long-standing open question whether $C_p(X)$ has such a quotient for any infinite compact space $X$. The above concept as well as the last problem are closely related to Efimov's problem that has remained open for 40 years. We will discuss a number of conditions that imply 2DCP. For example, every locally homogeneous compact space, every space containing a copy of $\beta\omega$ or $2^\omega$ has 2DCP although compact $h$-homogeneous spaces with 2DCP without such copies exist in ZFC. We prove that no scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces $X$ the 2DCP is equivalent to uncountability of $X$. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mr\'owka compacta. We give positive classes among zero-dimensional compact spaces; for example, the Brech, as well as the Sobota-Zdomskyy compact spaces of Efimov type have 2DCP. Open questions are included.

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math.GN 2026-06-25

Quotient maps by ideals preserve topological MV-algebra structure

by Li-Hong Xie, Jiang Yang

Quotient homomorphisms of Topological MV-Algebras and Applications

The map is always continuous and open; when the ideal is compact it is perfect and three-space theorems follow for compactness and related p

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For a topological group, the quotient map modulo a subgroup is open and the quotient map modulo a compact subgroup is perfect. In this paper we prove and develop the corresponding compact-ideal theory for topological $MV$-algebras. We show that if $I$ is an ideal of a topological $MV$-algebra $A$, then the natural quotient homomorphism $q:A\longrightarrow A/I$, where $A/I$ is endowed with the quotient topology, is a continuous open quotient map and $A/I$ is again a topological $MV$-algebra. If, in addition, $I$ is compact, then $q$ is perfect. As applications, we study three-space phenomena in topological $MV$-algebras. Under compact-kernel hypotheses we prove three-space theorems for compactness, local compactness, $\sigma$-compactness, Lindel\"ofness and paracompactness under the separation hypotheses stated below. We also prove a first-countability three-space theorem for locally convex topological $MV$-algebras.

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math.GN 2026-06-25

Continuously homogeneous hereditarily indecomposable continua are tree-like

by Jan Boroński, David Prier +2 more

The result extends homeomorphism theorems to continuous surjections and partially addresses whether such spaces must be homogeneous.

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A topological space $X$ is continuously homogeneous if for any $x,y\in X$ there exists a continuous surjection $f:X\to X$ with $f(x)=y$. We show that continuously homogeneous hereditarily indecomposable continua are tree-like, therefore, extending results of Bing and Rogers for homeomorphism and a result of Sturm for the pseudo-circle and pseudo-solenoids. This also provides a partial answer to the question of Lewis whether all continuously homogeneous hereditarily indecomposable continua are homogeneous.

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math.FA 2026-06-25

Metric sphere isometries reduce to homeomorphisms

by Katsuhisa Koshino

Isometries between the unit spheres of spaces of metrics

For non-degenerate compact metrizable spaces, every surjective isometry on the unit sphere of bounded pseudometrics is induced by a homeomor

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Given a topological space $Z$, let $PM(Z)$ be the space of bounded continuous pseudometrics on $Z$, which is endowed with the sup-norm, and let $SPM(Z)$ be the unit sphere of $PM(Z)$. In this paper, we shall prove that for all non-degenerate compact metrizable spaces $X$ and $Y$, and for any surjective isometry $T : SPM(X) \to SPM(Y)$, there exists a homeomorphism $\phi : Y \to X$ such that for any metric $d \in SPM(X)$ and for any pair of points $(x,y) \in Y^2$, $T(d)(x,y) = d(\phi(x),\phi(y))$. As a corollary, we can solve a variant of Tingley's problem on spaces of metics.

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math.GN 2026-06-24

Roelcke and WAP compactifications described for cyclic order groups

by Georgii Sorin

Roelcke and WAP compactifications of automorphism groups of ultrahomogeneous cyclically ordered sets

Explicit descriptions are given for automorphism groups of discrete ultrahomogeneous cyclically ordered sets in the pointwise convergence to

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In this work we describe Roelcke and WAP compactifications of automorphism groups of discrete ultrahomogeneous cyclically ordered sets in the topology of pointwise convergence.

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cond-mat.dis-nn 2026-06-24

Topological defects mark plastic sites in glasses

by Arabinda Bera, Peter Schall +3 more

The Physics of Topological Defects in Glasses

Invariants in modes and displacements correlate with soft spots and shear bands, offering a new view of amorphous yielding.

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Topological defects play a central role in the mechanical behavior of crystalline materials, yet their relevance to amorphous solids has only recently begun to emerge. Over the last few years, theoretical, computational, and experimental studies have revealed the presence of well-defined topological invariants in vibrational eigenmodes, non-affine displacement fields, and deformation-induced vector fields of glasses. These defects have been shown to correlate strongly with soft spots, localized plastic rearrangements, yielding, and shear-band formation, suggesting a new perspective on the microscopic origins of plasticity in disordered materials. In this review, we provide a comprehensive overview of recent developments in the rapidly growing field of topological defects in glasses. We discuss the underlying theoretical concepts, including Burgers vectors, non-affine plasticity, vibrational modes, and topological invariants, and review recent numerical and experimental advances. Finally, we assess the current achievements, limitations, and open questions, and discuss future directions toward a unified topological description of plasticity and mechanical failure in amorphous solids.

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math.CO 2026-06-24

Euler characteristic of parallel toric complements reduces to subtori

by Elia Saini

On the Euler-Poincar\'e characteristic of parallel toric arrangements

Basic cohomology equates the full complement invariant to a combination drawn from the singular subtori.

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Toric arrangements of maximal rank have been studied by the author in a paper that shows how the complement manifold of these arrangements is diffeomorphic to that of centered ones. In this work we turn our attention to toric arrangements of rank one, namely parallel toric arrangements. Our aim is to prove, by means of basic arguments of cohomology theory, that the Euler-Poincar\'{e} characteristic of the complement manifold of parallel toric arrangements can be computed in terms of those of the complement manifolds of the singular subtori that compose the arrangement.

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math.GN 2026-06-24

Semantic concepts formalized as product-bundle sections

by Jean-Philippe Garnier (Br.AI.K)

Fractal Algebraic Topology of Semantic Computation. A Peer-Review-Oriented Formalization of the SSTD/BrainiaK Concept Bundle

Elementary proofs establish continuity of operations on containers with R^14 sensor base and grammatical fibres.

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This manuscript develops material from the internal French research notes Traite de Topologie Algebrique Fractale into an academic manuscript. The editorial rule is strict: implementation names are not used as mathematical proofs, analogies are not promoted to theorems, and every formal result is either proved from explicit assumptions or downgraded to a model law, conjecture, or empirical claim. The central object is T n , a finite heterogeneous concept container formalized as a section of a product bundle whose slots include an empirical sensorimotor base R 14 , grammatical fibres, polarity, intensity, vision and audition slots, an SSTD spectral slot, a refined compositional fibre, and auxiliary tool/metric/axis/hint slots. We prove elementary structural results about product-bundle representation, heterogeneous GCM metrics, and continuity of componentwise operations. We then give conditional results for Frobenius-inspired crystal composition, Gamma/CNS curvature-Hopf modelling, Kalman convergence, SSTD bundle morphisms, and SpiderR flat-connection idealizations. Each conditional result includes its assumptions, proof status, implementation correspondence, and the boundary between mathematics, model assumptions, and empirical evidence.

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math.GN 2026-06-23

Clopen Boolean algebra left introverted only if group compact or discrete

by Joshua Basman Monterrubio, Thomas Czyzowicz +2 more

Totally Disconnected Semigroup Compactifications: Non-Introversion of the Full Boolean Algebra of Clopen Sets

Holds for first countable σ-compact totally disconnected locally compact groups and settles an earlier open question.

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In terms of the existence of a single clopen set and two related nets, we characterize when the full Boolean algebra, ${\mathfrak B}(G)$, of clopen subsets of a topological group $G$ is left introverted. We employ this characterization to show that when $G$ is a first countable, $\sigma$-compact, totally disconnected locally compact group, ${\mathfrak B}(G)$ is left introverted if and only if $G$ is compact or discrete, thus providing a strong positive answer to a question posed in Stephens and Stokke (Q J Math 2023). Examples of clopen sets and nets witnessing our non-introversion theorem are presented. Some hereditary properties of left introversion of ${\mathfrak B}(G)$ are proved and then employed to extend our main result to other classes of topological groups.

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math.GN 2026-06-23

Local finiteness equals closed images for all group cellular automata

by Jiang Yang

Closed Image Characterizations of Locally Finite Groups via Cellular Automata

The equivalence holds for any infinite alphabet and also for linear automata over infinite-dimensional vector spaces.

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We prove that a group $G$ is locally finite if and only if, for some (equivalently, every) infinite set $A$, every cellular automaton $A^G\to A^G$ has closed image in the prodiscrete topology. Equivalently, this holds if and only if every linear cellular automaton $V^G\to V^G$ has closed image for some pair $(K,V)$ with $V$ infinite-dimensional over the field $K$ (equivalently, for every such pair). This gives affirmative answers to Open Problems 6 and 7 of Ceccherini-Silberstein and Coornaert. More precisely, if $G$ is not locally finite, then for every infinite set $A$ there is a finite-memory cellular automaton $A^G\to A^G$ with non-closed image, and for every field $K$ and every infinite-dimensional $K$-vector space $V$ there is such a linear cellular automaton $V^G\to V^G$. The common obstruction is constructed on a countable direct-sum alphabet from an infinite ray in a locally finite Cayley graph. A direct-summand argument gives arbitrary vector-space alphabets, while an alphabet-retract principle gives arbitrary infinite set alphabets.

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math.GR 2026-06-22

Compact groups are Lie if free actions are all principal

by Alexandru Chirvasitu

Action principality as a Lie-group certificate

This holds for groups whose identity component has metrizable abelianization, as a converse to Gleason's theorem.

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A continuous action $\mathbb{G}\circlearrowright X$ of a topological group is principal if its isotropy groups are all conjugate to $\mathbb{H}\le \mathbb{G}$ and the quotient map $X\to X/\mathbb{G}$ is a locally trivial $\mathbb{G}/\mathbb{H}$-fiber bundle. We prove that compact groups whose identity component has metrizable abelianization are Lie provided their free actions on Tychonoff (equivalently, compact Hausdorff) spaces are all principal; this is a converse to Gleason's theorem. A variant confirms the conclusion for Tychonoff or compact Hausdorff actions with constant central isotropy by compact connected groups.

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math.GN 2026-06-19

Delaunay states yield matrix invariants for colored braids

by Illia E. Rohozhkin

Invariants of the Colored Braid Groupoid

A dynamical system of points with triangulation states defines a groupoid representation of ColB(n) and maps to GL matrices over Q and C.

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In this paper, a braid is regarded as a dynamical system of points in the plane. The states of this dynamical system are given by Delaunay triangulations. This construction makes it possible to define an abstract groupoid $\overset{{abc}}{\mathcal{G}^{4}_{n+3}}$, which gives a representation of the colored braid groupoid $\text{ColB}(n)$. We define homomorphisms ${f}_{n+3}:\overset{{abc}}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{Q})$ and ${f}'_{n+3}:\overset{{abc}}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{C})$, and describe an algorithm for computing the resulting invariants.

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math.GN 2026-06-18

Property Q makes d-spaces well-filtered

by Xiaoquan Xu

Some new results on well-filteredness of T₀-spaces

Smyth power spaces of T0-spaces always satisfy the same property, so the construction preserves well-filteredness.

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For a $T_0$-space $X$, let $Q (X)$ be the poset of nonempty compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if it satisfies the following two conditions: (1) $\wedge K$ exists for any $K\in Q(X)$, and (2) for any filtered family $\{K_d : d\in D\}\subseteq Q(X)$ and $x\in X$, if $\bigvee^{\uparrow}_{d\in D}\bigwedge K_d$ exists and $x\not\leq \bigvee^{\uparrow}_{d\in D}\bigwedge K_d$, then there is $\varphi\in \prod\limits_{d\in D}\!\!K_d$ and an upper bound $u$ of $\varphi(D)$ such that $x\not\leq u$. In this paper, we prove that every $d$-space with property Q is well-filtered and the Smyth power space of a $T_0$-space always has property Q. Hence the Smyth power construction preserves the well-filteredness. For a complete lattice $L$ and an order-compatible $d$-topology $\tau$ on it, we show that when $L$ possesses a certain distributivity, $(L, \tau)$ is well-filtered.

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math.DS 2026-06-15

Shadowing systems factor through surjective SFT inverse limits

by Dekui Peng

Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits

Surjective bonding maps force the Mittag-Leffler condition and let the zero-dimensional extension keep shadowing.

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Good and Meddaugh proved that every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type. We show first that, for this factor representation alone, both assumptions are unnecessary: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type. In particular, the mere existence of such a symbolic inverse-limit representation is not specific to shadowing. The main contribution of the paper is to identify the additional stability which shadowing provides in the metric case. We prove that every compact metric system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type whose bonding maps are surjective. Hence the inverse sequence satisfies the Mittag-Leffler condition, and the corresponding zero-dimensional extension still has shadowing. This strengthens the metric representation theorem of Good and Meddaugh and completes their characterization in terms of ALP factors of Mittag-Leffler inverse sequences of shifts of finite type. Finally, for arbitrary compact Hausdorff spaces, we show that every compact shadowing system is conjugate to the inverse limit of metrizable shadowing systems with factor bonding maps. In this sense, compact shadowing systems are generated from shifts of finite type by applying, at most three times, the two shadowing-preserving operations of taking Mittag-Leffler inverse limits and passing to ALP factors.

0

math.FA 2026-06-12

Consistently a small Grothendieck C(K) lacks convex renormings

by Todor Manev, Damian Sobota +1 more

A small Banach space C(K) without nice renormings

Under ω₁ < 𝔠 there is a compact K whose C(K) has density ω₁ but admits neither strictly convex nor sequentially Kadets-Klee renormings.

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We prove that consistently $\omega_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $\omega_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.

1 0

0

math.DS 2026-06-11

Continuum-wise hyperbolicity equals pseudo-Anosov with spine singularities

by Rodrigo Arruda, Bernardo Carvalho +2 more

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities

Such maps exist only on the torus and sphere and are conjugate to the standard examples there.

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We establish a complete structural classification for continuum-wise hyperbolic surface homeomorphisms. Specifically, we prove that a surface homeomorphism is cw$_F$-hyperbolic if, and only if, it is a pseudo-Anosov homeomorphism whose singularities consist exclusively of spines (1-prongs). Furthermore, we classify these systems up to topological conjugacy, showing that every such homeomorphism is conjugate to either an Anosov automorphism on the torus $\mathbb{T}^2$ or to its standard hyperelliptic quotient on the sphere $\mathbb{S}^2$. As a rigid consequence of this classification, we show that such dynamics are strictly obstructed on surfaces of genus greater than one, the Klein bottle, and the projective plane.

0

math.AT 2026-06-10

Weighted topologies interpolate between Hausdorff and final on Ran spaces

by Sylvain Douteau, Marie Labeye

Old and new structures on Ran spaces: Length structures, completeness, and conicality

They equip the final topology with a complete uniformity and yield conical stratification when the base is Riemannian.

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We study topologies on Ran spaces. In the literature, two distinct topologies frequently appear: the Hausdorff topology, and a finer one constructed as a colimit, that we call the final topology. In this work, given a metric space $M$, we construct new metric topologies on $\mathrm{Ran}(M)$, called weighted topologies. They interpolate between the Hausdorff and final topologies, the later being recovered as a limit in the category of spaces. This structure equips the final topology with a uniformity, which we show to be complete. Finally we study the Ran spaces as stratified spaces. We show that, whenever $M$ is a Riemannian manifold, the weighted topologies are conically stratified, while the final topology is only so in a weak sense.

0

math.OA 2026-06-09

C*-simple groups have exactly one amenable recurrent subalgebra

by Tattwamasi Amrutam, Pierre Fima +1 more

Uniformly recurrent subalgebras in finite von Neumann algebras

In crossed products with amenable coefficients, a group is C*-simple precisely when the only amenable uniformly recurrent subalgebra contain

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We introduce the notion of a uniformly recurrent subalgebra (URA) for a trace-preserving action of a countable discrete group $\Gamma$ on a finite von Neumann algebra $M$, providing an operator-algebraic counterpart to the theory of uniformly recurrent subgroups (URS). We also show that the Effros-Mar\'echal space $\text{Sub}(M)$ is compact if and only if $M$ lacks a diffuse direct summand. Leveraging this, we show that URAs can exhibit arbitrary topological complexity and construct exotic URAs homeomorphic to any prescribed minimal Polish space. In the context of crossed products $M \rtimes \Gamma$ with amenable coefficients, we utilize URAs to formulate a new characterization of C*-simplicity, proving that $\Gamma$ is C*-simple if and only if the only amenable URA of the crossed product containing $M$ is $\{M\}$. Finally, to bypass the failure of compactness in $\text{Sub}(M)$, we develop a generalized state-space machinery using Baire-category methods on the weak-* compact space of trace-extending states. This construction captures compact, discrete, and exotic URAs, while recovering the classical URS framework as a special case.

0

math.GN 2026-06-09

Topological MV-algebras are Mal'tsev spaces

by Li-Hong Xie, Jiang Yang

Pseudocompact Topological $MV$-Algebras

This lets arbitrary products of pseudocompact examples stay pseudocompact.

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Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we find that every topological $MV$-algebra is a Mal'tsev space introduced by Mal'tsev in 1954. Hence, applying the theorem of Reznichenko and Uspenskij on pseudocompact Mal'tsev spaces, we show that the product of arbitrary family of pseudocompact topological $MV$-algebras are pseudocompact. We also prove that every $\sigma$-compact topological $MV$-algebra is ccc. Secondly, we obtain that the Stone-\v{C}ech compactification of a pseudocompact topological $MV$-algebra carries a natural compact topological $MV$-algebra structure extending the original one. Finally, we prove that: let $I$ be a closed ideal in a pseudocompact topological $MV$-algebra $A$ and $\iota_1:A\hookrightarrow\bet A$ is the naturally injective; then $\cl_{\beta A}\iota_1(I)$ is a closed ideal of $\beta A$ and $ \beta A/\cl_{\beta A}\iota_1(I)\cong \beta(A/I)$.

0

math.GN 2026-06-08

Every locally compact Hausdorff space is a maximal d-spectrum

by G. Bezhanishvili, P. Bhattacharjee +2 more

Maximal d-spectra and locally compact Hausdorff spaces

Generalizing the d-nucleus via Priestley duality realizes them from continuous regular frames, plus a corollary for locally Stone spaces.

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It is an interesting open problem whether every compact Hausdorff space can be realized as the maximal $d$-spectrum of an arithmetic frame. We approach this problem by generalizing the $d$-nucleus to a stably continuous frame. We use Priestley duality to characterize the resulting $\underline d$-nucleus, which allows us to prove that every locally compact Hausdorff space can be realized as the maximal $\underline d$-spectrum of a continuous regular frame. As a corollary, we obtain that every locally Stone space can be realized as the maximal $d$-spectrum of an algebraic regular frame.

0

math.GN 2026-06-08

Hausdorff MV-algebras embed into pathwise connected versions

by Li-Hong Xie, Jiang Yang

The Hartman--Mycielski construction in topological MV-algebras

The construction embeds Hausdorff MV-algebras into pathwise connected ones as closed subalgebras while extending maps continuously.

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Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we mainly show that for every Hausdorff topological MV-algebra $A$, there exists a natural topological isomorphism $i_A:A\rightarrow A^\bullet$ of $A$ onto a closed subalgebra of the pathwise connected, locally pathwise connected topological MV-algebra $A^\bullet$. Furthermore, we show that there is an extension to a bounded continuous function on the MV-algebra $A^\bullet$ for each continuous real-valued bounded function on a topological MV-algebra $A$. Finally, we prove that if $\varphi:A_1\rightarrow A_2$ is a continuous homomorphism of topological MV-algebras, then $\varphi$ admits a natural extension to a continuous homomorphism $\varphi^\bullet:A_1^\bullet\rightarrow A_2^\bullet$; in addition, if $\varphi$ is open and onto, then so is $\varphi^\bullet$.

0

math.GN 2026-06-08

Ultrafilters split five t-norms into four classes near 1

by Jiang Yang, Xiongwei Zhang +1 more

Ultrafilter Equivalence and Asymptotic Types of Five Classical t-Norms

Three operations reduce to the zero operation in the low-value regime, with quotient category and ultrapower interpretations.

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We study five classical $t$-norms on the unit interval from the viewpoint of ultrafilter concentration. For a fixed ultrafilter $\mathcal U$ on $[0,1]$, we introduce an equivalence relation identifying two operations whenever they coincide on $A\times A$ for some $A\in\mathcal U$. We show that their asymptotic behavior is governed by two concentration regimes. In the near-$1$ regime, the five operations determine four distinct ultrafilter-equivalence classes. In the low-value regime, the {\L}ukasiewicz, nilpotent minimum, and drastic $t$-norms collapse to the zero operation. We encode these reductions in a discrete quotient category and record simple ultrametric models for the two regimes. We further interpret the classification inside classical ultrapowers: the near-$1$ and near-$0$ regimes become exact algebraic phenomena on infinitesimal monads, and saturation yields a compactness principle for countable systems of asymptotic identities. Finally, we indicate how the same viewpoint interacts with residual fuzzy implications generated by $t$-norms.

0

math.GN 2026-06-08

Convex powerdomains reflect continuity and quasicontinuity but do not preserve the latter

by Yuxu Chen

Directed Convex Powerspaces and Convex Powerdomains

Upper and convex cases now complete the full preservation and reflection profile across all powerdomains on dcpos.

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It is known that lower powerdomains preserve and reflect both continuity and quasicontinuity, while the preservation of quasicontinuity by upper and convex powerdomains had long been open. Directed spaces provide a topological extension framework for dcpos. Powerdomains of dcpos can be characterized as the $D$-completions of the corresponding directed powerspaces. Using this observation, the authors proved in 2024 that upper powerdomains do not preserve quasicontinuity. In this paper, we prove that directed convex powerspaces and convex powerdomains do not preserve quasicontinuity, but they do reflect both continuity and quasicontinuity. We also prove that upper powerspaces and upper powerdomains reflect quasicontinuity. Together with the known results for lower and upper powerdomains, these arguments give the complete preservation and reflection profile of the lower, upper and convex powerdomains for continuity and quasicontinuity.

0

math.GN 2026-06-08

AM-GM inequality defines new contractions with fixed points

by Irom Shashikanta Singh, Yumnam Mahendra Singh +2 more

Arithmetic-geometric mean, additive, and multiplicative contractions: New generalizations of the Banach contraction principle

Continuous mappings obeying the new conditions on spaces with auxiliary semimetric δ possess fixed points; regularity on δ makes the results

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We introduce new contraction conditions based on classical inequality between arithmetic and geometric means. By incorporating an auxiliary semimetric $\delta$, we define arithmetic-geometric mean, multiplicative-type, and additive-type contractions. Connections between these types of contractions are found. Fixed point theorems are proved in the case of continuity of the above mentioned contractions. Under suitable regularity conditions on $\delta$ (such as being d-regular, strongly d-regular, or d-lower bounded) we obtain constructive corollaries. Various examples demonstrating our results are constructed. It is shown that with certain caveats fixed point theorem for additive-type mappings is equivalent to the fixed point theorem for perturbed metric spaces, which were recently introduced by M. Jleli and B. Samet.

0

math.GN 2026-06-05

Metric dimension equals density character in non-archimedean fields

by S. Maghsoudi, Daniel L. Rodríguez-Vidanes

Two notes on valued fields

Multiplication is uniformly open on every valued field in a way that depends on the metric and not just the topology.

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This paper studies two questions on valued fields: the metric dimension induced by an absolute value, and the uniform openness of multiplication. For nontrivial non-archimedean absolute values, we prove that the metric dimension equals the density character. In the archimedean case, the metric dimension is 2 for subfields of $\mathbb{R}$, while for non-real subfields of $\mathbb{C}$ it is either 2 or 3, according to whether the field contains a non-real element together with its complex conjugate. We also show that multiplication is uniformly open on every valued field. Finally, we prove that this property is genuinely metric, not purely topological, even on $\mathbb{R}$ with a suitable compatible choice of metric.

0

math.GN 2026-06-05

MT-algebras yield choice-free Wallman compactifications

by Sebastian D Melzer, Cerene Rathilal +1 more

A choice-free approach to Wallman compactifications

Working with powerset MT-algebras instead of spaces allows Wallman and Stone-Čech compactifications to be built without choice until spatial

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The classical Wallman compactification of a $T_1$-space and the Stone--\v{C}ech compactification of a completely regular space rely on choice principles. We show that, by representing a space by its powerset MT-algebra (McKinsey--Tarski algebra), both constructions admit choice-free compactifications. More generally, from any Wallman basis of a spatial $T_1$ MT-algebra we construct a compact $T_1$ MT-algebra which is a compactification of the original algebra. Taking the basis of all closed elements yields a choice-free Wallman compactification of every spatial $T_1$ MT-algebra, while taking the basis of zero-elements yields a choice-free Stone--\v{C}ech compactification of every spatial completely regular MT-algebra. Choice is only needed to show that the resulting compactifying algebras are spatial, and hence to recover the usual compactifying spaces. We also show that these constructions recover the corresponding compactifications of frames of opens.

0

math.GN 2026-06-03

Cyclic orders decide when 1D maps extend to plane homeomorphisms

by Dinh Si Tiep, Nhan Nguyen

Planar extensions in o-minimal structures

Necessary and sufficient combinatorial conditions on orders and orientations guarantee definable extensions from closed 1-dimensional sets.

abstract click to expand

Let $X \subset \mathbb{R}^2$ be a closed definable set of dimension at most $1$, and let $h : X \to \mathbb{R}^2$ be a definable continuous injective map. In this paper, we establish necessary and sufficient combinatorial conditions, formulated in terms of cyclic orders at topological singular points and orientations of Jordan curves, for $h$ to admit a definable homeomorphic extension to the whole plane.

0

math.DS 2026-06-02

Stationary spaces admit only 0

by Arie Levit, Kfir Silman

Ends of stationary metric measure spaces

Random metric measure spaces including graphs and manifolds restricted by return-time analysis

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We prove that stationary random metric measure spaces have 0,1,2 or a Cantor space of ends. This notion includes stationary random graphs, manifolds and discrete subgroups. In the case of surfaces, we classify all possible homeomorphism types, in analogy with the work of Biringer and Raimbault on unimodular Riemannian manifolds. Our approach relies on a general "no geometric core" principle and an analysis of finite versus infinite expected return times.

0

math.GN 2026-06-01

Continuous hyperspace selections pick endpoints in connected spaces

by Valentin Gutev

On the Variety of Hyperspace Selections

The set of all such points has a totally disconnected closure and is closed inside first countable totally disconnected spaces.

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If $f$ is a continuous selection for the Vietoris hyperspace $\mathcal{F}(X)$ of the nonempty closed subsets of a space $X$, then the point $p=f(X)\in X$ is not as arbitrary as it might seem at first glance. In fact, the set $\mathcal{O}_{cs}(X)$ of all these points reveals certain information about the variety of Vietoris continuous selections for $\mathcal{F}(X)$. Thus, for a connected space $X$, we will show that every point $p\in \mathcal{O}_{cs}(X)$ is not only noncut, but also an endpoint of $X$. Another result of this paper is that in an arbitrary topological space $X$, the closure of the set $\mathcal{O}_{cs}(X)$ is always a totally disconnected subset. Furthermore, we will also show that $\mathcal{O}_{cs}(X)$ is a closed subset of every first countable totally disconnected space $X$.

0

math.OA 2026-06-01

Every [F,F]-invariant subalgebra of L(F) equals L(N) for normal N

by Tattwamasi Amrutam, Artem Dudko

Relative invariant subalgebra rigidity for Thompson's group F

The result follows from a general factoriality criterion that applies once F is shown to be i.c.c., simple, and to have only essentially fre

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We prove that Thompson's group $F$ satisfies the relative invariant subalgebra rigidity property with respect to its commutator subgroup: every von Neumann subalgebra of $L(F)$ that is invariant under conjugation by $[F,F]$ is of the form $L(N)$ for some normal subgroup $N \trianglelefteq F$. Along the way, we establish a general factoriality criterion for invariant subalgebras whose hypotheses are met whenever the ambient group is i.c.c., simple, and every faithful ergodic measure-preserving action of it on a probability space is essentially free.

0

math.GT 2026-06-01

MCS spaces are CS sets with identical stratifications

by Mohammad Alattar, Lewis Tadman

MCS Spaces are CS

The intrinsic stratification matches the MCS one, strengthening Perelman's result and resolving Fujioka's question.

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In this paper we further develop the theory of MCS spaces. Our main result shows that MCS spaces, as defined by Perelman, are CS sets with respect to their MCS stratification, and that in fact, the intrinsic stratification agrees with the MCS stratification. As a consequence, we improve on Perelman's result and answer affirmatively a question by Fujioka.

0

math.GT 2026-06-01

Typical isotopies fix projection dimensions of compact sets in R^N

by Olga Frolkina

On projections of a compact set in mathbb R^N

The pattern yields new tests to decide whether a Cantor set is tame or wild by examining its shadows after a generic move.

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We apply ideas of geometric measure theory and Baire category theory to topological problems, namely, to topological embeddings of compact sets into Euclidean spaces. In 1947, Borsuk constructed a Cantor set in $\mathbb R^N$, $N\geqslant 3$, such that its projection onto any $(N-1)$-plane contains an $(N-1)$-dimensional ball. This can be strengthened: a desired Cantor set can be obtained from an arbitrary Cantor set by an arbitrarily small isotopy of the space $\mathbb R^N$. The question arises: how do the dimensions of the projections of a compact set $X\subset \mathbb R^N$ behave under a typical ambient isotopy or under a typical ambient homeomorphism? (Typical in the sense of the Baire category.) We solve this problem. As a consequence, we get new criteria of tameness and wildness of a Cantor set in terms of its projections. Our main result strengthens V{\"a}is\"{a}l\"{a}'s theorem (1979) connecting Hausdorff dimension and Shtan'ko embedding dimension. In its turn, V{\"a}is\"{a}l\"{a}'s theorem extends results of N\"{o}beling (1931) and Szpilrajn (1937) on relationship between Hausdorff dimension and topological dimension.

0

math.GN 2026-05-29

σ-compact paratopological groups may have only the trivial group as continuous image

by Pedro J. Chocano, Tayomara Borsich

On continuous isomorphisms from σ-compact paratopological groups onto topological groups

Explicit examples show continuous isomorphisms exist only when the target topology is trivial, answering an open question in the negative.

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In this short note we give a negative answer to the following open question: \emph{Let $X$ be a $\sigma$-compact paratopological group. Does there exist a continuous isomorphism of $X$ onto a topological group $G$?} Specifically, we construct a family of $\sigma$-compact paratopological groups for which a continuous isomorphism onto a topological group $G$ exists if and only if $G$ carries the trivial topology.

0

math.GR 2026-05-27

Infinite-image endomorphisms on N admit only pointwise Polish topology

by Serhii Bardyla, Luna Elliott

Polish topologies on endomorphism monoids of linear orders

Full monoids End(N,≤) and End(Z,≤) have infinitely many, while End(N,<) has exactly continuum many and lacks a maximal second-countable topo

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In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which yields automatic continuity of Borel measurable homomorphisms between certain topological semigroups. This provides a new method for analyzing Polish semigroup topologies on monoids with small groups of units. We show that for all monoids considered, the semigroup Zariski topology coincides with the pointwise topology and is therefore the coarsest Hausdorff semigroup topology. We prove that the submonoid $\operatorname{End}^{\infty}(\mathbb{N},\leq)$ of $\operatorname{End}(\mathbb{N},\leq)$ consisting of all endomorphisms with infinite image admits a unique Polish semigroup topology, namely the pointwise topology. On the other hand, despite possessing a finest Polish semigroup topology, the monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$, admit infinitely many distinct Polish semigroup topologies. Also, we show that the monoid $\operatorname{End}(\mathbb{N},<)$ admits exactly $2^{\aleph_0}$ Polish semigroup topologies and no maximal second-countable semigroup topology.

0

math.GN 2026-05-26

Free groups on compact uniform spaces match U^ω in cofinal type

by Xuan Gong, Dekui Peng

Cofinal types of topological groups

Neat trees refine covering trees to turn character equalities into full Tukey equivalences.

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We investigate the local topological structure of non-metrizable topological groups through the lens of Tukey order and cofinal types. Motivated by recent advances in topological groups admitting an $\omega^\omega$-base, we introduce the \emph{fineness index}, denoted $\f(P)$, for arbitrary directed partially ordered sets. This cardinal invariant fundamentally generalizes the bounding number $\mathfrak{b}$ by capturing the exact threshold where a poset evades domination by its countable subsets, thereby establishing a universal lower bound for the character of topological groups with a $P$-base: $\chi(G) \in \{1, \omega\} \cup [fi(P), \text{cof}(P)]$. Furthermore, we resolve a structural problem regarding the exact cofinal types of free topological groups over uniform spaces. While classical results by Nickolas, Tkachenko, and others successfully computed the character of these groups via cardinal equalities (e.g., $\chi(F(X, \mathcal{U})) = \text{cof}(\mathcal{U}^\omega)$), lifting these equalities to strict Tukey equivalences has remained a persistent combinatorial challenge. By developing the novel machinery of \emph{neat trees} to refine uniform covering trees, we overcome the structural obstructions and prove the Tukey equivalence $\Ne_e(F(X, \U))=_T \U^\omega$ for any compact uniform space $(X, \U)$.

0

math.GN 2026-05-25

Mapping spaces C(X,Y) are G-UANR exactly when Y is UA(N)R

by Sergey A. Antonyan, Luis A. Martínez-Sánchez

Equivariant homotopy dense subsets in the realm of uniform G-ANR spaces

The equivalence follows from G-homotopy dense subsets characterizing G-ANRs and applies to all finite-group Lawson metric G-semilattices.

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Let $G$ be a compact group. The existence of certain $G$-homotopy dense subsets in a metrizable $G$-space $X$ plays a fundamental role, as it is equivalent to $X$ being a $G$-ANR. From this perspective, the present paper develops several applications of this class of $G$-subsets. In particular, we prove that for a compact $G$-space $X$ and a metric space $Y$, the mapping space $C(X,Y)$ is a $G$-UA(N)R if and only if $Y$ is a UA(N)R in the sense of Michael. This result is significant because it enables the construction of examples of Lawson metric $G$-semilattices for which the property of being a $G$-UANR is equivalent to uniform local path-connectedness. Moreover, we show that this equivalence holds for every Lawson metric $G$-semilattice whenever $G$ is finite. Finally, we analyze the behavior of $G$-homotopy dense subsets when the ambient space is a $G$-A(N)R, thereby introducing the notion of a $G$-A(N)R-pair.

0

math.GR 2026-05-25 Recognition

Left coarse structures on quotients do not always match

by Carlos Pérez Estrada, Christian Rosendal

Coarse Structures on Homogeneous Spaces

Mapping class groups of Loch Ness monster surfaces give a counterexample where the structure on G/H differs from the quotient structure on G

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Given a closed normal subgroup $H$ of a topological group $G$, we address the question of whether the left coarse structure on the quotient group $G/H$ equals the quotient of the left coarse structure on $G$. We provide a counterexample among Polish groups, namely, the mapping class group of the Loch Ness monster surface seen as a quotient of the mapping class group of the punctured Loch Ness monster surface, and establish both equivalent and sufficient conditions for when this holds in special settings. The latter are formulated in terms of liftings of bounded sets, existence of transversals and metrisability of the left coarse structure of $G$ restricted to $H$.

0

math.GN 2026-05-25 Recognition

Order-n graphic contraction forces a periodic point

by Evgeniy Petrov

Periodic point theorem for generalized graphic contractions

The inequality on n-th iterates guarantees some x returns to itself after finite steps of T.

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Let $(X,d)$ be a nonempty metric space and let $n\in \mathbb N^+$. We shall say that $T\colon X\to X$ is a graphic contraction of order $n$ if there exists $\alpha\in (0,1)$ such that the inequality $$ d(T^n x,T^{2n}x) \leqslant \alpha d(x,T^nx) $$ holds for all $x\in X$. In the case $n=1$ these mapping are known as graphic contractions and are well studied. In the present paper, we establish a theorem on the existence of periodic points for a graphic contraction of order $n$. Examples of such mappings having different properties are constructed.

0

math.GN 2026-05-25 Recognition

Three separability notions coincide for σ-compact gyrogroups

by Shumin Lai, Fucai Lin

The separability embedding of σ-compact strongly topological gyrogroups

Homeomorphic embedding into a separable regular space is equivalent to two gyrogroup embeddings when the gyrogroup is σ-compact.

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In this paper, it is shown that every right $\omega$-narrow strongly topological gyrogroup $G$ is right $\omega$-balanced by applying the gyrosemidirect product groups. Then we investigate the class of $\sigma$-compact strongly topological gyrogroups, and conclude that every $\sigma$-compact strongly topological gyrogroup is range-metrizable. By applying these results, we discuss the separability embedding of $\sigma$-compact strongly topological gyrogroups, and claim that the following three statements (a)-(c) are equivalent for any $\sigma$-compact strongly topological gyrogroup $G$: \smallskip (a) $G$ is homeomorphic to a subspace of a separable regular space; \smallskip (b) $G$ is topologically gyrogroup isomorphic to a subgyrogroup of a separable strongly topological gyrogroup; \smallskip (c) $G$ is topologically gyrogroup isomorphic to a closed subgyrogroup of a separable path-connected, locally path-connected strongly topological gyrogroup. The above results extend the classical results from topological groups to the class of strongly topological gyrogroups in the literature.

0

math.GN 2026-05-22 Recognition

Reflective subcategories extend to presheaf categories

by Julio César Hernández Arzusa, Hernán Giraldo +1 more

Reflections and Sheafifications in Algebraic and Topological Categories

If C reflects in A then presheaves over C reflect in presheaves over A, and reflections match sheafifications under natural conditions.

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In this article, we develop an explicit categorical realization of sheafification based on colimits, products, and subobjects, emphasizing its behavior in algebraic and topological-algebraic settings. We prove that if $\mathcal{C}$ is a reflective subcategory of a category $\mathcal{A}$, then the presheaf category $\mathbf{Psh}(X,\mathcal{C})$ is reflective in $\mathbf{Psh}(X,\mathcal{A})$. We further investigate the interaction between reflections and sheafification, obtaining natural conditions under which these constructions are naturally isomorphic.

0

hep-th 2026-05-22 1 theorem

Huge representations of U_q(su_N) yield shaded A-polynomials for SU(N)

by Dmitry Galakhov, Alexei Morozov

Shading A-polynomials via huge representations of U_q(mathfrak{su}_N)

Double scaling limit converts Clebsch-Gordan chords into classical constraints on SU(N) character varieties of knot complements.

abstract click to expand

Classical A-polynomials $A(\ell,m)$ define constraints on coordinates $\ell$ and $m$ in $SL(2,\mathbb{C})$ (a complexification of $SU(2)$) character varieties associated to knot complements $S^3\setminus K$. Quantum A-polynomials $\hat A(\hat \ell,\hat m)$ are difference operators annihilating Jones polynomials believed to represent wave functions of 3d Chern-Simons theory with gauge group $SU(2)$ on a toroidal pipe surrounding the knot $K$ strand -- a boundary of the knot complements $S^3\setminus K$. We suggest a construction of classical shaded A-polynomials $A_a(\ell_b,m_c)$ associated to Lie groups $SU(N)$. We exploit a formalism of Clebsh-Gordan (CG) chords, where indices $a$, $b$, $c$ run over $1,\ldots,N-1$. CG chords have a natural interpretation in terms of 2d CFTs of WZW type, or, alternatively, in terms of quantum group $U_q(\mathfrak{su}_N)$. In the case of $\mathfrak{su}_2$ CG chords could be associated to Reeb chords in a knot contact homology (KCH) framework. KCH suggests its own analogue of A-polynomials known as augmentation polynomials allowed to have extra spurious roots in principle. Yet the CG chord formalism could be easily extended to arbitrary $\mathfrak{su}_N$ allowing us to generalize the construction of A(ugmentation)-polynomials to arbitrary $\mathfrak{su}_N$ and arbitrary representation as well. Primarily we aim at classical A-polynomials by considering a double scaling limit when $q=e^{\hbar}$, $\hbar\to 0$ and the representations are huge, in particular, highest weight vector components $w_i\to \infty$ so that $\hbar w_i\sim m_i$ remain finite. Still we expect the presented techniques would be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras $\mathfrak{g}$. Also we discuss explicit examples of A-polynomials for knots $3_1$, $4_1$ and $5_1$ for $\mathfrak{g}=\mathfrak{su}_3$.

1 0

0

math.CT 2026-05-20 Recognition

Localic groupoids classify strictly more logical theories than toposes

by Graham Manuell, Joshua L. Wrigley

Generic bundles over a localic category

They recover geometric theories with stronger universal properties and classify dual theories for proper separated bundles.

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In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories, demonstrating that these groupoids satisfy a stronger universal property than that of their corresponding classifying toposes. We also prove a dual result that there exist classifying localic categories and groupoids for proper separated bundles satisfying a dual geometric theory. Thus, localic groupoids classify strictly more kinds of logical theories than toposes. Our approach provides a concrete construction of the localic categories and the generic bundles involved in terms of generalised frame presentations. To accommodate our approach, we prove en passant a constructive, pointfree version of the Alexandroff--Hausdorff theorem and that internal functors that are fully faithful and effective descent morphisms on objects induce equivalences between the categories of discrete opfibrations over the source and target categories.

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math.RT 2026-05-20 1 theorem

Graph function lattices match quiver subrepresentation lattices

by Giovanni Cerulli Irelli, Domenico Fiorenza +2 more

A categorification of Kauffman states for planar graphs

Under suitable assumptions, ω-compatible angular functions on planar graphs yield graded distributive lattices isomorphic to maximal quiver-

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Given a decorated planar graph $(G,\omega)$, where $G$ is a planar graph and $\omega\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $\omega$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,\omega)$, which is the directed graph of such functions, and $BMS(G,\omega)$, the directed graph of BMS states which are some pairs of $\omega$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,\omega)$ to be a graded distributive lattice, recovering Kauffman's Clock Theorem when $G$ is a knot diagram. We also define a potential on $\mathcal{Q} G$ and associate a representation of the corresponding quiver with potential to every BMS state. Under suitable assumptions, this construction yields an isomorphism between $\mathcal{L}(G,\omega)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte--Schiffler.

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eess.SP 2026-05-20 Recognition

Neighborhood selection cuts indoor positioning errors

by Neetu R. R, Shrihari Vasudevan +1 more

Measurement Selection Strategies for Position Estimation in Indoor Environments

Ray-tracing maps access point groups to pick reliable measurements and reduce non-line-of-sight delays in dense spaces.

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Time-based indoor positioning techniques rely on multiple access points (APs) and measurements between the user equipment (UE) and the APs. In dense indoor environments, occlusion-induced non-line-of-sight (NLoS) propagation introduces significant delays in these measurements, thereby degrading position estimation accuracy. To address this challenge, this paper proposes measurement selection strategies to improve position estimation accuracy. A ray-tracing (RT) simulator is employed to characterize the propagation environment and derive AP neighborhood information, which is subsequently used to design and evaluate different measurement selection strategies. The approaches explored include AP neighborhood-based cardinality selection, intersection and union of measurements from AP neighborhoods, and fixed measurement selection. Experiments demonstrate the efficacy of the proposed measurement selection strategies in environments under significant NLoS conditions.

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math.GN 2026-05-19 1 theorem

Completeness alone fails to make uniform spaces Baire

by Francisco Javier García Pacheco, Álvaro García Zambrano +1 more

The Baire property in uniform spaces: a survey

Survey maps conditions like countable compactness and pseudocompleteness that restore the property automatic in the pseudometric case.

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The Baire category theorem states that every complete pseudometric space is a Baire space. There are some results in metric spaces which have their analogue in uniform spaces, however this is not one of them. Nonetheless, since the Baire property is always desirable, we decide to explore some conditions, such as countable compacity, pseudocompacity and pseudocompleteness, and see in which circumstances a general complete uniform space satisfies the Baire property.

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math.GN 2026-05-18

Infinite elementary Abelian groups always have infinite MAP subgroups

by Ol'ga Sipacheva

Maximally almost periodic subgroups of elementary Abelian topological groups

The result holds for every topology making the group operations continuous.

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It is proved that any infinite elementary Abelian topological group has an infinite maximally almost periodic subgroup.

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math.GN 2026-05-18 2 theorems

New uniformity on median algebras yields minimal compactification

by Michael Megrelishvili

Intrinsic uniform structure on median algebras

For finite intervals it matches the Roller compactification and makes finite-rank group systems dynamically tame.

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We introduce the median uniformity $\mathcal U_{\mathrm m}$, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology $\tau_{\mathrm m}$ provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When all intervals in the median algebra $X$ are finite, the MMC is the unique proper median compactification of $(X,\tau_{\mathrm m})$; in particular, it coincides with the Roller compactification. We apply this uniform framework to continuous actions of a topological group $G$ by median automorphisms. We show that the MMC is a median $G$-compactification. In the finite-rank case, the resulting compact $G$-system is Rosenthal representable and hence dynamically tame.

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math.GN 2026-05-15

Biquandle quivers produce persistent homology invariants for links

by Hamdi Kayaslan

Persistent Homology of Biquandle Coloring Quivers

Extending clique complexes to quivers and filtering endomorphism sets yields invariants unchanged by Reidemeister moves.

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In this paper, we extend the notion of directed clique complex to quivers and introduce an associated homology theory. By applying this construction to biquandle coloring quivers, we obtain new invariants of links. We then introduce a quiver filtration-valued invariant of links induced by filtrations of biquandle endomorphism sets. We construct persistent homological invariants of links by applying persistence techniques to these quiver filtrations through the introduced homology theory.

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math.AG 2026-05-13

Reeb spaces of continuous functions reduce to one-dimensional graphs

by Naoki Kitazawa

Representations of Reeb spaces via simplified graphs and examples

Nice Hausdorff domains yield 1D Reeb spaces representable by simplified graphs even when not CW complexes.

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Reeb spaces of continuous real-valued functions on topological spaces are fundamental and strong tools in investigating the spaces. The Reeb space is the natural quotient space of the space of the domain represented by connected components of its level sets. They have appeared in theory of Morse functions in the last century and as important topological objects, they are shown to be graphs for tame functions on (compact) manifolds such as Morse(-Bott) functions and naturally generalized ones. Related general theory develops actively, recently, mainly by Gelbukh and Saeki. For nice Haudorff spaces and continuous functions there, they are "$1$-dimensional". We concentrate on Reeb spaces which are not CW complexes and study their representations by graphs and nice examples. Reconstructing nice smooth functions with given Reeb graphs is of related studies and pioneered by Sharko and followed by Masumoto, Michalak, Saeki, and so on. The author has also contributed to it.

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math.CO 2026-05-13 Recognition

Minimal sequence collections detect discontinuities

by Gyuhyun Lim

On minimal collections of sequences for testing continuity

Test sets smaller than all convergent sequences suffice under natural hypotheses at P.

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We study test sets: subfamilies of sequences converging to a point P that still suffice to detect every discontinuity of real-valued functions at P. Ordered by inclusion, these test sets form a poset. Under natural hypotheses at P, we prove that this poset has a minimal element. We also analyze its maximal chains, showing that some have a least element, while others do not. Finally, on the sequential fan we give a concrete realization in which the minimal test set produced by our construction has strictly smaller cardinality than the full family of convergent sequences.

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math.LO 2026-05-12 Recognition

ZFC proves adp equals dp

by Vinicius de Oliveira Rodrigues

Almost Disjointness Principles and Q-Space Cardinals

The almost disjointness principle matches the dominating number in ZFC, while its tree version at can be forced strictly larger than ap.

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Banakh and Bazylevych introduced separation-axiom variants $\mathfrak q_i$, for $i=1,2,2\frac{1}{2}$, of the cardinal $\mathfrak q$, together with a cardinal $\mathfrak{adp}$ lying between $\mathfrak{dp}$ and $\mathfrak{ap}$. They asked whether $\mathfrak{adp}$ coincides with either of these two cardinals. We prove in ZFC that $\mathfrak{adp}=\mathfrak{dp}$. We define a dual variant $\mathfrak{adp}_2$ and show that $\mathfrak{adp}_2=\mathfrak{ap}$. We further study the relation between $\mathfrak{ap}$ and the weakened $Q$-space cardinals. We introduce a tree analogue $\mathfrak{at}$ of $\mathfrak{ap}$ and prove $\mathfrak q_1\leq\mathfrak{at}\leq\mathfrak q_{2\frac{1}{2}}$, hence $\mathfrak{ap}\leq\mathfrak q_{2\frac{1}{2}}$. Assuming the Generalized Continuum Hypothesis, we construct ccc forcing extensions with $\mathfrak{ap}=\omega_1<\mathfrak{at}=\mathfrak q_{2\frac{1}{2}}=\mathfrak c$, so $\mathfrak{ap}<\mathfrak{at}$ is consistent with ZFC.

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math.FA 2026-05-12 Recognition

Equivariant subhomogeneous C* bundles are pullbacks from compact spaces

by Alexandru Chirvasitu

Obstructed subhomogeneous-bundle extensions and embeddings

Finite-type ones on normal spaces are locally trivial vector bundles or arise from universal compactifications or maps to smooth manifolds

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We address a number of problems concerning the (im)possibility of either extending locally trivial subbundles of possibly singular Banach/$C^*$ bundles globally, embedding subhomogeneous bundles into homogeneous ones, or recovering locally trivial compact-Lie-group-equivariant Banach or $C^*$ bundles as pullbacks along equivariant maps to compact spaces. The results include (1) the global extensibility of a locally trivial Banach/Hilbert/Banach-algebra/$C^*$ subbundle from a closed subspace of a paracompact space given appropriate homotopy constraints; (2) the homogeneous embeddability of equivariant subhomogeneous Banach/Hilbert bundles locally trivial along the singular locus under the same homotopy constraints, and (3) the characterization of finite-type equivariant locally trivial subhomogeneous $C^*$ bundles on normal spaces as precisely those (a) locally trivial as plain vector bundles, or (b) pulled back from the universal equivariant compactification or (c) pulled back from an equivariant map into a smooth manifold. The latter extends results of Phillips concerning non-equivariant matrix-algebra bundles restricted along the Stone\v{C}ech compactification.

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cs.LO 2026-05-12 1 theorem

Every prevision is the infimum of sublinear previsions

by Jean Goubault-Larrecq

Just Previsions

The same holds as a supremum of superlinear ones under mild conditions, yielding homeomorphisms to hyperspaces via orthogonality.

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Previsions are positively homogeneous functionals, and are generalized forms of integration functionals. We investigate previsions -- just previsions, not sublinear or superlinear previsions as in previous work. We show that every prevision can be expressed as an infimum of sublinear previsions, and as a supremum of superlinear previsions under mild conditions. This extends to homeomorphisms between spaces of previsions and certain hyperspaces over spaces of sublinear or superlinear previsions, which can also be characterized in terms of orthogonality relations, making the construction a variant of a double powerspace construction.

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cs.FL 2026-05-11 2 theorems

Asymptotic Hausdorff turns edit distances into language metrics

by Dana Fisman, Gal Meirom

Asymptotic Hausdorff and Language Similarity

The construction ignores finite mismatches and yields computable distances between regular languages based on their long-word behavior.

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We introduce the \textit{Asymptotic Hausdorff} lifting, denoted $\mathbb{AH}_{d}$, a general method for lifting an element-level metric $d$ to a (pseudo-) metric on sets, that captures asymptotic similarity in infinite domains equipped with a notion of size. The construction is designed to be insensitive to finite deviations and to avoid the limitations of classical Hausdorff-based approaches, which are often overly sensitive to outliers and fail to reflect asymptotic behavior. Formal languages provide a central motivating instance of this framework, where elements are words and sets are languages. When applied to normalized edit distances, the Asymptotic Hausdorff lifting yields metric-valued distances between languages that reflect asymptotic edit behavior while preserving metric structure. We study the equivalence classes of regular languages induced by $\mathbb{AH}_{d}$ for normalized edit distances $d$, and characterize their asymptotic essence. Focusing in particular on the normalized edit distance of Marzal and Vidal, $\textsf{ned}$, we investigate the computation of $\mathbb{AH}_{\textsf{ned}}$ for regular languages and for bounded context-free languages.

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math.LO 2026-05-11

Box topologies define meager ideals for singular cardinals

by Yusuke Hayashi, Tristan van der Vlugt

Topology and category for singular product spaces

Candidate higher Baire and Cantor spaces allow cardinal characteristics of κ-meager sets to be studied when κ is singular.

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For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where $\kappa$ is singular. We will consider several spaces of functions and box topologies that could serve as higher Baire and Cantor spaces for singular cardinals. The ultimate focus of the article lies in studying cardinal characteristics of the ideal of $\kappa$-meagre subsets of these spaces.

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math.LO 2026-05-11 2 theorems

Ultrafilter quotients characterize self-divisible ultrafilters

by Manoranjan Singha, Rohan Pradhan

Reply to Some Questions of Quotients when ultrafilters divide ultrafilters

When v strongly divides u the defined u/v yields stability for idempotents and characterizes multiplicative delta sets.

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For ultrafilters u,v on N, the operation u/v is introduced and formalised which acts as quotient-like structures when v strongly divides u.Central to our study is the characterization of self-divisible ultrafilters in connection with the divisibility of u by u/v.Some results on the algebraic stability of multiplicative udempotents are presented.The paper also connects the combinatorial notions such as multiplicative delta sets,provoding characterization via self-divisible ultrafilters.

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math.GN 2026-05-11 Recognition

GSI2-convergence is topological exactly in strongly QI2-continuous T0-spaces

by Xinpeng Wen, Meng Bao +1 more

On GSI2-convergence in T0-spaces

The equivalence holds for every irreducible complete T0-space, giving a direct link between a new convergence and a continuity property.

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In this paper,we introduce the concept of GSI$_2$-convergence in $T_0$ spaces and the related concept of (strongly) QI$_2$-continuous spaces. It is proved that if GSI$_2$-convergence in $X$ is topological iff $X$ is strongly QI$_2$-continuous for any irreducible complete $T_0$ space $X$.

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math.GR 2026-05-11 2 theorems

If additive is solvable, so is multiplicative in connected skew braces

by Marco Damele, Andrea Loi

Solvability and Rigidity for Topological Skew Braces

The implication holds for connected locally compact Hausdorff topological skew braces, with counterexamples arising when any condition isdro

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We study compact and locally compact topological analogues of the Byott--Vendramin solvability problem for finite skew braces, asking whether solvability of the additive group forces solvability of the multiplicative group. Our main theorem proves an affirmative result in the connected locally compact Hausdorff setting: if $B=(B,\cdot,\circ)$ is a connected locally compact Hausdorff topological skew brace and the additive group $(B,\cdot)$ is solvable, then the multiplicative group $(B,\circ)$ is solvable. The proof proceeds by reducing the additive group to a solvable Lie quotient and then applying an affine-action theorem: a connected Lie group acting transitively and affinely on a connected solvable Lie group, with solvable stabilizer identity component, is itself solvable. We further show that the Hausdorff, local compactness, and connectedness hypotheses are essential by constructing counterexamples when each is omitted. In the compact connected Hausdorff case with abelian additive group, we obtain a stronger rigidity phenomenon: the two group laws coincide.

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math.GN 2026-05-08 1 theorem

Graph intersections fix Nielsen bound for multi-valued map coincidences

by Grzegorz Graff, P. Christopher Staecker +1 more

Nielsen coincidence theory of (n,m)-valued pairs of maps

Corrected invariant gives sharp lower bound on coincidence points for n- and m-valued maps on the circle.

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We consider pairs of maps $(f,g)$, where $f$ is an $n$-valued map and $g$ is an $m$-valued map, defined on connected finite polyhedra. A point $x$ such that $f(x)\cap g(x)\neq \emptyset$ is called a coincidence point of $f$ and $g$. A useful device for studying coincidence points would be a Nielsen-type invariant which provides a lower bound for the number of coincidence points of all $(n, m)$-valued pairs of maps homotopic to $(f,g)$. The construction of such an invariant $N(f:g)$ was proposed in [J. Fixed Point Theory Appl. 14, 309--324 (2013)]. Unfortunately, this approach has some flaws. In this paper, we present a modified construction that yields a corrected form of the invariant, defined in terms of the intersection points of the graphs of $f$ and $g$. In the case of $(n, m)$-valued pairs of maps of the circle our invariant provides a sharp lower bound, which we precisely determine.

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math.AG 2026-05-08

QHD degenerations of elliptic surfaces receive complete classification

by Marcos Canedo, Giancarlo Urzúa

Rational homology disk degenerations of elliptic surfaces

Extending Kawamata, the work realizes all cases on Dolgachev surfaces and constructs unobstructed minimal models that connect to Lee-Lee

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In this paper, a $\mathbb{Q}$HD singularity is a weighted homogeneous normal surface singularity admitting a rational homology disk ($\mathbb{Q}$HD) smoothing. These singularities are rational but often not log canonical. We classify all $\mathbb{Q}$HD degenerations of nonsingular projective elliptic surfaces, extending Kawamata's classification of the case with only Wahl singularities (i.e., log terminal $\mathbb{Q}$HD singularities). We also realize all $\mathbb{Q}$HD degenerations of Dolgachev surfaces $D_{a,b}$ with one $\mathbb{Q}$HD singularity, for every pair of integers $a,b$. For each such degeneration, we construct a minimal semi log canonical (slc) birational model via a Seifert partial resolution in the sense of Wahl followed by semistable flips. Finally, we prove that these minimal slc models are unobstructed and deform to the recent degenerations of Dolgachev surfaces constructed by D. Lee and Y. Lee.

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math.GN 2026-05-06

Transitive distributive binary G-spaces classified for compact G

by Pavel S. Gevorgyan

On the Transitive Binary G-Spaces

Subgroups from distributive sets stay distributive and a criterion is given, with full classification for compact groups.

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Distributive subsets of the group of all invertible continuous binary operations on a topological space are considered, and it is proved that the subgroups generated by them are also distributive. A criterion for the distributivity of a binary action of a topological group $G$ on a space $X$ is obtained. The concept of transitive binary $G$-space is introduced, and a classification of transitive distributive binary $G$-spaces is given in the case of a compact group $G$.

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math.GR 2026-05-06 3 theorems

Z-groups pass finite entropy to closed derived subgroups

by Sonia L'Innocente, Francesco G. Russo +1 more

A dynamical approach to Schur's Theorem

A dynamical reading of Schur's theorem shows that topological entropy on continuous endomorphisms is inherited by the commutator subgroup in

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A classical result of Schur of 1904 shows that an infinite (discrete) group $E$ with finite central quotient $E/Z(E)$ should have finite derived subgroup $[E,E]$. Schur's Theorem has many important consequences, which have been extensively investigated in the literature. Here we focus on topological Hausdorff groups, which are not necessarily discrete groups, and show a dynamical version of Schur's Theorem via the notion of topological entropy of Adler, Konheim and McAndrew. Their perspective follows some original intuitions of Kolmogov and Sinai from the area of the dynamical systems. Firstly, we investigate the topological entropy of continuous endomorphisms of maximal almost periodic groups whose closed derived subgroup is compact. The properties of these groups were known to Takahashi in 1952 and among them we find the $\mathsf{Z}$-groups of Grosser and Moskowitz. Secondly, we give a new dynamical interpretation of the Schur's Theorem, showing that a $\mathsf{Z}$-group $G$ with continuous endomorphisms of finite topological entropy should have closed derived subgroup $\overline{[G,G]}$ with continuous endomorphisms of finite topological entropy. Finally, we illustrate a series of constructions and examples, which allow us to justify our interpretation of Schur's Theorem as generalization of the original version.

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math.NT 2026-05-06

Lectures recast topology as condensed sets for algebraic use

by Peter Scholze

Lectures on Condensed Mathematics

The notes explain how to replace ordinary spaces with sheaves on profinite sets so limits and maps behave algebraically.

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This is an updated version of the lectures notes for a course on condensed mathematics taught in the summer term 2019 at the University of Bonn. The material presented is joint work with Dustin Clausen. This is intended as a stable citable version of the original lectures, with mostly cosmetic changes to the original document, together with some small corrections.

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math.GN 2026-05-04

Duality theorem connects binary group actions to topological fields

by Pavel S. Gevorgyan

Binary transformation groups and topological fields

Semitransitive distributive binary G-spaces correspond to topological fields with multiplicative group G, giving category equivalence andnew

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The notion of a semitransitive binary action of a group $G$ on a topological space is introduced. A duality theorem is proved, establishing a bijective correspondence between semitransitive distributive binary $G$-spaces and topological fields whose multiplicative group is isomorphic to $G$. This result yields an equivalence between the category of semitransitive distributive binary $G$-spaces and the category of topological fields with multiplicative group $G$. As applications of the duality theorem, two important results are established. It is shown that a finite group can act semitransitively, distributively, and binarily only on finite sets whose cardinality is a power of a prime number. A complete characterization of those groups that can appear as multiplicative groups of topological fields is also obtained.

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math.GN 2026-05-04 1 theorem

Coarser metrics preserve perfect maps between paracompact spaces

by Vlad Smolin

Perfect maps between submetrizable spaces

A positive answer shows that perfectness survives when choosing metrizable coarsenings compatible with the map.

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We investigate a question posed by Huaipeng Chen: if $X$ and $Y$ are paracompact submetrizable spaces and $f:X\to Y$ is a perfect map, can $X$ and $Y$ be submetrized by metrics $\rho$ and $d$ respectively such that $f$ remains perfect with respect to the induced topologies?

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math.GN 2026-05-01

Countable centered local π-bases bound ccc Hausdorff spaces to size 𝔠

by Nathan Carlson

On centered local π-bases

This improves the Hajnal-Juhász theorem by weakening first-countability to the centered local π-base condition.

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In 1967 Hajnal and Juh{\'a}sz showed that the cardinality of a first-countable Hausdorff space with the countable chain condition has cardinality at most $\mathfrak{c}$, the cardinality of the real line. We give an improvement of this celebrated theorem by replacing ``first-countable" with the weaker condition ``each point has a countable centered local $\pi$-base". Given a point $p$ in a topological space $X$, a \emph{local} $\pi$-\emph{base} $\scr{B}$ at $p$ acts like a neighborhood base at $p$ except that $p$ may not be in any member of $\scr{B}$. A local $\pi$-base $\scr{B}$ has the \emph{finite intersection property} if any finite intersection of members of $\scr{B}$ is nonempty. We call this type of local $\pi$-base \emph{centered}. A centered local $\pi$-base behaves even more like a neighborhood base in a sense. A space has the \emph{countable chain condition} if every family of pairwise disjoint open sets is countable. We also improve a theorem of Pospi{\v s}il from 1937 using centered local $\pi$-bases. As is customary, examples are given to demonstrate these improvements are strict. Compact Hausdorff spaces are also explored in this connection, along with variations on the notion of a centered local $\pi$-base.

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