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Good Fibrations through the Modal PrismAug 21 2019Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected points of ... More On the growth rate of dichromatic numbers of finite subdigraphsAug 20 2019Chris Lambie-Hanson proved recently that for every function $ f:\mathbb{N}\rightarrow \mathbb{N} $ there is an $ \aleph_1 $-chromatic graph $ G $ of size $ 2^{\aleph_1} $ such that every $ (n+3) $-chromatic subgraph of $ G $ has at least $ f(n) $ vertices. ... More Directed Homotopy in Non-Positively Curved SpacesAug 19 2019A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of ... More Effectivizing Lusin's TheoremAug 17 2019Lusin's Theorem states that, for every Borel-measurable function $\bf{f}$ on $\mathbb R$ and every $\epsilon>0$, there exists a continuous function $\bf{g}$ on $\mathbb R$ which is equal to $\bf{f}$ except on a set of measure $<\epsilon$. We give a proof ... More Cardinal invariants of Haar null and Haar meager setsAug 15 2019Aug 19 2019A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $\mu$ and a Borel set $B$ containing $X$ such that $\mu(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric ... More Cardinal invariants of Haar null and Haar meager setsAug 15 2019A ssubset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $\mu$ and a Borel set $B$ containing $X$ such that $\mu(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric ... More Lifting recursive counterexamples to higher-order arithmeticAug 15 2019In classical computability theory, a recursive counterexample to a theorem shows that the latter does not hold when restricted to computable objects. These counterexamples are highly useful in the Reverse Mathematics program, where the aim of the latter ... More Bootstraps, nets, and hierarchiesAug 15 2019Classification is at the heart of the scientific enterprise, from bacteria in biology to groups in mathematics. A central classification project in mathematical logic is motivated by Goedel's incompleteness theorems. Indeed, logical systems are classified ... More Towers and clubsAug 14 2019We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal on a regular ... More Properties of the connective implication in effect algebrasAug 14 2019Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serves as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, ... More The Power of the Weisfeiler-Leman Algorithm to Decompose GraphsAug 14 2019The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited ... More Modelling socio-political competitionAug 13 2019This paper continues the investigation of the logic of competing theories, be they scientific, social, political etc. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive graphs, inspired ... More The logic of vague categoriesAug 13 2019We introduce a complete many-valued semantics for basic normal lattice-based modal logic. This relational semantics is grounded on many-valued formal contexts from Formal Concept Analysis. We discuss an interpretation and possible applications of this ... More Polarized relations at singulars over successorsAug 13 2019Erdos, Hajnal and Rado asked whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_1}_2$. We prove that both relations ... More Piece selection and cardinal arithmeticAug 12 2019We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if $\lambda \geq 2^\kappa$, then (a) $I_{\kappa, \lambda}$ is not $(\lambda, ... More Representing Polish groupoids via metric structuresAug 08 2019We prove that every open $\sigma$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{\omega_1\omega}$-sentence in continuous logic. In particular, the orbit equivalence relations ... More On Extensions of Partial IsomorphismsAug 08 2019In this paper we study a notion of HL-extension (HL standing for Herwig--Lascar) for a structure in a finite relational language $\mathcal{L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal{L}$-structure. In addition, ... More Almost Indiscernible Theories and Saturated Free AlgebrasAug 07 2019We extend the concept of "almost indiscernible theory" introduced by Pillay and Sklinos in arXiv:1409.8604 (which was itself a modernization and expansion of Baldwin and Shelah (Algebra Universalis, 1983)), to uncountable languages and uncountable parameter ... More On double-membership graphs of models of Anti-FoundationAug 07 2019We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected ... More A Note on the "Third Life of Quantum Logic"Aug 07 2019The purpose of this note is to discuss some of the questions raised by Dunn, J. Michael; Moss, Lawrence S.; Wang, Zhenghan in Editors' introduction: the third life of quantum logic: quantum logic inspired by quantum computing. A Note on the Possibility of Self-Reference in MathematicsAug 07 2019In this paper we propose an interpretation for self-referential propositions in a "meta-model" N* of ZF. This meta-model N* is considered as an informal model of arithmetic that mathematicians often use when working with number theory. Specifically, we ... More A game-theoretic proof of Shelah's theorem on labeled treesAug 07 2019We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality $\kappa$ of the family is much larger (in the sense of large cardinals) than the cardinality $\lambda$ of the set of labels, more precisely ... More Superstability, noetherian rings and pure-semisimple ringsAug 06 2019We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We obtain a characterization of noetherian rings via superstability of the class of left modules with embeddings. $\mathbf{Theorem.}$ ... More Pseudo-finite sets, pseudo-o-minimalityAug 05 2019We give an example of a model of the common theory of o-minimal structures, in a given language, and an expansion of that model admitting the exact same one-variable definable sets with the latter admitting a definable, closed, bounded, and discrete subset ... More Chain Logic and Shelah's Infinitary LogicAug 03 2019For a cardinal of the form $\kappa=\beth_\kappa$, Shelah's logic $L^1_\kappa$ has a characterisation as the maximal logic above $\bigcup_{\lambda<\kappa} L_{\lambda, \omega}$ satisfying Strong Undefinability of Well Order (SUDWO). SUDWO is a strengthening ... More A study of truth predicates in matrix semanticsAug 01 2019Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic L is associated with a matrix semantics Mod*(L). This paper is a contribution to the systematic ... More On the complexity of the Leibniz hierarchyAug 01 2019We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. ... More A computational glimpse at the Leibniz and Frege hierarchiesAug 01 2019In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem ... More Epimorphism surjectivity in varieties of Heyting algebrasAug 01 2019It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by ... More Bounds on Continuous Scott RankAug 01 2019An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathscr{L}$ be a language of continuous logic with code $\hat{\mathscr{L}}$. Let $\Omega$ ... More A refinement of the Ramsey hierarchy via indescribabilityJul 31 2019Aug 13 2019A subset $X$ of a cardinal $\kappa$ is Ramsey if for every partition $f:[X]^{<\omega}\to 2$ there is a set $H\subseteq X$ of cardinality $\kappa$ which is \emph{homogeneous} for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<\omega$. ... More A refinement of the Ramsey hierarchy via indescribabilityJul 31 2019A subset $X$ of a cardinal $\kappa$ is Ramsey if for every partition $f:[X]^{<\omega}\to 2$ there is a set $H\subseteq X$ of cardinality $\kappa$ which is \emph{homogeneous} for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<\omega$. ... More Complexity in Young's LatticeJul 31 2019We investigate the complexity of the partial order relation of Young's lattice. The definable relations are characterized by establishing the maximal definability property modulo the single automorphism given by conjugation; consequently, as an ordered ... More Ultrafilters on singular cardinals of uncountable cofinalityJul 26 2019We prove that consistently there is a singular cardinal $\kappa$ of uncountable cofinality such that $2^\kappa$ is weakly inaccessible, and every regular cardinal strictly between $\kappa$ and $2^\kappa$ is the character of some uniform ultrafilter on ... More Equivalence à la Mundici for lattice-ordered monoidsJul 26 2019We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MMV-algebras (for `Monoidal MV-algebras'). ... More Simultaneously vanishing higher derived limitsJul 26 2019In 1988, Sibe Marde\v{s}i\'{c} and Andrei Prasolov isolated an inverse system $\mathbf{A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim^n\mathbf{A}$ ... More Extensional Higher-Order Paramodulation in Leo-IIIJul 26 2019Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple ... More The spectrum problem for Abelian l-groups and MV-algebrasJul 25 2019Aug 05 2019This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian l-groups. As a first main result, we show that a topological space $X$ is the prime spectrum of an MV-algebra ... More The spectrum problem for Abelian l-groups and MV-algebrasJul 25 2019This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian l-groups. As a first main result, we show that a topological space $X$ is the prime spectrum of an MV-algebra ... More Uniform Martin's conjecture, locallyJul 24 2019We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol x \le_T \boldsymbol ... More Semigroups, Projections, $κ$-domainsJul 24 2019We propose an extension of the classical notion of projection to semigroups and provide conditions under which a semigroup embeds in a complete lattice. We also introduce the new notion of a $\kappa$-domain and prove some useful separation results valid ... More Free Kleene algebras with domainJul 24 2019First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the ... More The word problem for double categoriesJul 23 2019We solve the word problem for double categories by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation is of interest in its own right since ... More The word problem for double categoriesJul 23 2019Aug 18 2019We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation ... More The word problem for double categoriesJul 23 2019Jul 28 2019We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation ... More Solution of Equations Involving the Modular $j$ FunctionJul 23 2019Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular ... More Solutions of equations involving the modular $j$ functionJul 23 2019Aug 15 2019Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular ... More Splitting chains, tunnels and twisted sumsJul 23 2019We study splitting chains in $\mathscr{P}(\omega)$, that is, families of subsets of $\omega$ which are linearly ordered by $\subseteq^*$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{ZFC}$. We show that they ... More The $k$-Dimensional Weisfeiler-Leman AlgorithmJul 22 2019In this note, we provide details of the $k$-dimensional Weisfeiler-Leman Algorithm and its analysis from Immerman-Lander (1990). In particular, we present an optimized version of the algorithm that runs in time $O(n^{k+1}\log n)$, where $k$ is fixed (not ... More Examples of weak amalgamation classesJul 22 2019We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite ... More Resetting $α$-register machines and ZF$^{-}$Jul 22 2019We study the computational strength of resetting $\alpha$-register machines, a model introduced by P. Koepke. Specifically, we prove the following strengthening of another recent result of ours: For an exponentially closed ordinal $\alpha$, we have $L_{\alpha}\models$ZF$^{-}$ ... More Monomialization of a quasianalytic morphismJul 22 2019We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes (of interest in real analysis), the class of infinitely differentiable functions ... More Partitioning a reflecting stationary setJul 19 2019We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer that it is ... More Provenance Analysis for Logic and GamesJul 19 2019A model checking computation checks whether a given logical sentence is true in a given finite structure. Provenance analysis abstracts from such a computation mathematical information on how the result depends on the atomic data that describe the structure. ... More Override and updateJul 19 2019Override and update are natural constructions for combining partial functions, which arise in various program specification contexts. We use an unexpected connection with combinatorial geometry to provide a complete finite system of equational axioms ... More Rediscovered theorem of LuzinJul 18 2019In 1934 N. N. Luzin proved in his short (but dense) paper \textit{Sur la decomposition des ensembles} that every set $X\subseteq \mathbb{R}$ can be decomposed into two full, with respect to Lebesgue measure or category, subsets. We will try to (at least ... More A note on the classification of Gamma factorsJul 18 2019One of the earliest invariants introduced in the study of finite von Neumann algebras is the property Gamma of Murray and von Neumann. In this note we prove that it is not possible to classify separable $\rm{II}_1$ factors satisfying the property Gamma ... More Degrees of Randomized ComputabilityJul 17 2019In this survey we discuss work of Levin and V'yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V'yugin introduced an ordering ... More 19th century real analysis, forward and backwardJul 17 201919th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in ... More On mathematical realism and the applicability of hyperrealsJul 16 2019Jul 30 2019We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between ... More On mathematical realism and the applicability of hyperrealsJul 16 2019We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between ... More A Canonical Model Proof of Strong Completeness for BQLJul 16 2019I prove using a canonical model construction that a simple extension of Visser's natural deduction system for Basic Propositional Logic is both sound and strongly complete with respect to Basic First-Order Logic (BQL). I utilize the canonical model construction ... More Time-Stamped Claim LogicJul 15 2019Jul 16 2019The main objective of this paper is to define a logic for reasoning about distributed time-stamped claims. Such a logic is interesting for theoretical reasons, i.e., as a logic per se, but also because it has a number of practical applications, in particular ... More A note on local $ω$-consistency and reflectionJul 15 2019In this note we answer a question of R. Kaye and H. Kotlarski regarding the relationship between the schemata of local $\omega$-consistency $\omega\textsf{-}\mathsf{Con}(\mathsf{PA})$ and local reflection $\mathsf{Rfn}(\mathsf{PA})$ for Peano arithmetic ... More Ordinal sums of triangular norms on a bounded latticeJul 15 2019The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit interval states ... More