15,879 papers in this slice of arXiv.
Peter Banáš
Let X be a Polish space and let K(X) be its Vietoris hyperspace. A family I⊆K(X)
Artem Chernikov
The classes of SOP2 and SOP3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.
Mohammad Golshani
Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of σ-centered forcing notions gives a model of MAω1(σ-centered)+2ℵ0=ℵ2+ω12↛(ω12,3)2.
Naoya Kato
PAsmu− is a weak theory of arithmetic obtained by adding two principles concerning powers of two to basic axioms satisfied by the nonnegative part of a discretely ordered ring. We introduce a theory PAwit− in which powers of two and the required witnesses are represented by primitive symbols. We show that, along a fixed sequence in the standard model of arithmetic, every one-variable term eventually agrees with a polynomial over the dyadic rationals. A finite avoidance lemma and the Compactness Theorem then yield a model of PAsmu− in which division by 3 fails. In this model even the n=3 instance of the Standard Euclidean Division Principle pa16 fails. In fact, the model can be chosen to satisfy every universal L0-sentence true in the standard model. Consequently, neither pa16 nor the Euclidean Division Principle pa17 is derivable even after these universal truths are adjoined to PAsmu−.
Jonathan Feigert, Daniel Herden, Marcos Mazari-Armida
We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid S being LO (for every s,t∈S, we have that s∈St or t∈Ss) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when S is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when S is LO.
Jakub Rydval
We study the amalgamation decision problem: given a universal first-order sentence Φ, decide whether the class fm(Φ) of its finite models has the amalgamation property. We call Φ semantic Horn if fm(Φ) is closed under binary direct products. By McKinsey's theorem, this is equivalent to Φ being logically equivalent to a universal Horn sentence; the distinction is one of input representation, since Φ itself need not be given in Horn form and conversion to an explicit Horn normal form can incur an exponential blow-up. We prove that the problem is decidable under this semantic promise. Moreover, it belongs to 2EXPTIME, and to EXPTIME for every fixed bound on the arity of the input signature. Thus the semantic Horn fragment admits an unconditional decision procedure for signatures of unbounded relational arity. Our proof starts from the inside-out correspondence, which we use as a black box for the semantic reduction to a finite completion problem. We encode finite completions as homomorphisms to a finite relational template and introduce a finite set-valued local-consistency certificate for completion problems whose template has bounded width. For semantic Horn inputs, the local completions over each fixed source chart are closed under relationwise intersection of the added relations, and these intersections are compatible with restriction maps. This yields a semilattice polymorphism of the completion template. Since the template is binary, the semilattice operation gives width 2, making the local-consistency certificate complete. The same construction gives a decision procedure whenever the associated completion template has bounded width.
Tan Özalp
Answering a question of Todorcevic, we prove higher-dimensional canonization theorems for the topological Ramsey space FIN2[∞]. Our results build upon the work of Lopez-Abad and continue the line of research initiated by Erdős and Rado, and further developed by Pudlák and Rödl, Prömel and Voigt, Taylor, and Klein and Spinas. We identify the canonical functions on fronts of FIN2[∞] and develop an extension of the separating-mixing technique of Prömel and Voigt to canonize arbitrary functions g:F→ω, where F is a front of FIN2[∞]. We further extend our canonization theorem to arbitrary Borel maps g:FIN2[∞]→R, establishing new canonical Ramsey theorems for Polish spaces. In particular, we provide a complete classification of the canonical functions on FIN2[n], together with an explicit formula for their number as a function of n.
Sergio Celani, Paula Menchón, Valeria Miguelez
Kleene triples, introduced by Jalali, provide a representation of arbitrary Kleene algebras in terms of twist-products. We introduce modal Kleene algebras and modal Kleene triples, and establish a categorical equivalence between the corresponding categories, extending Jalali's duality to the modal setting. We study the centered case and show that Kleene algebras with implication can be naturally described within this framework by viewing implication as a family of modal operators. Finally, we develop a topological duality for modal Kleene triples.
Eugene Zhang
In this paper, a new model for multi-valued logic is presented based on the notion of a bundle of predicates. The degree of truth and various logical operations for predicates in our model of multi-valued logic are rigorously defined and investigated. Furthermore, the bundle of sets as a special type of predicate bundles is thoroughly investigated to provide a rigorous model for fuzzy sets and certain adjectives/adverbs in linguistics. In addition, solutions to necessity/possibility and the sorites paradox in modal logic are given.
Tomasz Kania
The stability programme initiated by Ulam asks when approximate solutions to algebraic identities must lie near exact ones. For lattices, this leads to the question of when a map that nearly preserves joins and meets can be approximated by a genuine lattice homomorphism. Badora--Kochanek--Przebieracz developed a neighbourhood-based framework for distributive lattices, centred on a separation (sandwich) lemma that constructs an exact join homomorphism between a join-subhomomorphism and a join-superhomomorphism via an order envelope. We revisit this mechanism and identify the single step at which distributivity is used: a decomposition identity for elements lying below a join. Without distributivity, separation can fail already in the modular lattice M3 and in a small finite orthomodular lattice. On the positive side, we show that separation holds in arbitrary lattices whenever the lower bounding map is isotone. For orthomodular lattices---algebraic models of quantum logic---we develop a blockwise stability theory on Boolean blocks. Approximate identities on compatible pairs yield exact homomorphic selections on each block (for joins, for meets, and for both operations under bi-admissibility). We present several gluing criteria for assembling blockwise selections, and we give a concrete finite example showing that gluing can fail when block overlaps are non-trivial. Finally, in the spirit of Kalton--Roberts, we obtain blockwise approximation results for nearly additive functions on orthomodular lattices by finitely additive measures, with an illustration on finite-dimensional projection lattices.
Yilong Zhang
We study the expansion of the real field by the graphs of power functions on the unit circle. Under a natural number-theoretic conjecture, we prove that adding such dense subsets does not increase the topological complexity of definable sets: every open definable set remains semialgebraic. The proof uses a two-sorted structure that separates the linear and algebraic data, inspired by Zilber's raising to powers. Using Hrushovski's amalgamation method, we construct and axiomatize a class of rich structures, and then show that the intended structure is a model. This provides a new example of a tame expansion of the real field by dense trajectories.
Haosui Duanmu, Xinyu Liu, David Schrittesser
Loeb measure theory stands as one of the most influential concepts in nonstandard analysis, underpinning nearly all applications in probability, stochastic processes, and mathematical economics. The paper resolves a fundamental open problem in Loeb measure theory originally posed by Keisler and Sun: let (Ω,F,μ) and (Ω,G,ν) be two Loeb equivalent internal probability spaces, and H be the internal algebra generated from F∪G. Does there exist an internal probability measure P on H such that (Ω,H,P) is Loeb equivalent to (Ω,F,μ)? While arXiv:2112.13955 recently provided a positive answer for hyperfinite probability spaces, the problem remained open for general internal probability spaces. We establish the existence of such an internal probability measure for all internal probability spaces.
Junhong Chen, Yi Zhang
We investigate the first-order theories of full n-branching ordinal trees Tαn. We obtain a complete classification of the standard trees up to elementary equivalence: every Tαn is elementarily equivalent to one of four canonical types determined by the ordinal α. For each canonical type we establish effective quantifier elimination and prove decidability of the theory. Along the way we develop the realizability theory of colored ordinal characters and obtain a sharp bound on the length of minimal witnesses. We also clarify the relationship between these trees and monadic second-order logic over ordinals, and show that the equal-height relation is not first-order definable in any standard tree.
Bernardo Subercaseaux, Benjamin Przybocki
Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the positive integers and yet does not follow from Tarski's axioms: align* &((1+x)^y + (1+x+x²)^y)^x · ((1+x³)^x + (1+x²+x⁴)^x)^y = &((1+x)^x + (1+x+x²)^x)^y · ((1+x³)^y + (1+x²+x⁴)^y)^x. align* Gurevič gave an algebra on 59 elements that satisfies Tarski's axioms but not Wilkie's identity, and over the years several authors whittled down the size of such a countermodel, culminating in a countermodel of size 12 due to Burris and Yeats. On the other hand, Zhang proved that there is no countermodel with fewer than 11 elements. Using SAT, we prove that the smallest countermodels are of size 12, as conjectured by Burris and Yeats. Moreover, we show that there are exactly 8,957,952 countermodels on 12 elements up to isomorphism and provide a simple classification of them. Our SAT approach outperforms dedicated tools for finding countermodels in equational theories, namely Mace4 and SEM. Furthermore, using autoformalization, we prove the correctness of our main result in Lean.
Martino Lupini
We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely Π1+λ+n+20 for a countable ordinal λ that is either zero or a limit ordinal, and a finite ordinal n. Considering Fréchet spaces produces the additional complexity classes Πλ0 for a countable limit ordinal λ.
Evgeny Kuznetsov
For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice K˚(Spec(R)) of compact open subsets of Spec(R) to be a Heyting algebra. Bezhanishvili and Tressl showed that K˚(Spec(C(T))) is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether Spec(C(βN)) is actually an Esakia space. We show it is not: applying our criterion to C(βN)≅ℓ∞(N,R) produces a diagonal counterexample, and the same obstruction rules out βD for every infinite discrete D. A grid-existence theorem for σ-complete Boolean algebras lets us push the construction to every basically disconnected compact Hausdorff space, settling the Bezhanishvili-Tressl question completely: for no infinite compact Hausdorff space T is Spec(C(T)) an Esakia space.
Silvan Horvath, Tan Özalp
We show that, up to isomorphism, the number of Q-points is either finite, 2d or 2c. This answers a question asked by Borodulin-Nadzieja, Martínez-Celis, Morawski and Świerczyńska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many Q-points implies the existence of non-atomic Q-measures, and of 2c-many Tukey-top Q-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..
Saugata Basu, Hamidreza Amini Khorasgani, Hemanta K. Maji +1
We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space. The right-hand sides are formal expressions generated from variables and parameters by convex linear combinations, finite union, and a positive geometric join encoding strict convex combinations. We prove that every parameter assignment has a unique smallest convex-set-valued solution and give a finite Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution and produces parameter-only expressions for its coordinate sets. More generally, let B be a family of subsets containing ∅ and closed under finite unions, nonnegative dilation, Minkowski sums, positive geometric joins, and convex hulls. If all parameter sets lie in B, then every coordinate set of the smallest solution lies in B; when these operations are effective, so is the resulting description. In particular, if the parameters are finite unions of hemihedra---where a hemihedron is a bounded convex semi-linear set, equivalently a convex finite union of relative interiors of polytopes---then each coordinate set is a hemihedron and admits a quantifier-free semi-linear description. We apply this theory to lamination hulls. For V=U⊕i=1⨁kWi,dimWi=1,Λ=i=1⋃k(U+Wi), we prove that the lamination hull GΛ(∞)(S) of every finite S⊂V is semi-algebraic and effectively computable by a quantifier-free formula over the reals.
Pedro Teixeira Yago
We study when embeddings lift from structures to their ultrapowers, and when an ultrapower embeds into its direct power. We offer a generalization of Blass's theorem on the Rudin-Keisler order and introduce cardinal invariants τA characterizing the embeddability AI/U↪AI. In infinitary languages, the criterion becomes necessary and sufficient: embeddability holds iff U is κ+-complete, when τAI/U=κ.
William Adkisson
The strong tree property and the super tree property (also called ITP) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal κ is strongly compact if and only if the strong tree property holds at κ, and supercompact if and only if ITP holds at κ. Generalizing a result of Golshani and Hayut, we show that from large cardinals it is consistent for ITP to hold simultaneously at any countable initial segment of successors of singular cardinals. More formally, given any countable ordinal θ, we construct a forcing extension in which ITP holds at the first θ successors of singulars. We then extend this result further to obtain the strong tree property on long segments of successors of singular cardinals of multiple cofinalities simultaneously.