Article
Mathematics
A. A. Stepanova
Summary: The paper focuses on injective S-acts with a P-stable theory and proves that the class of injective S-acts is (P, 1)-stable only if S is a one-element monoid. Additionally, it describes commutative and linearly ordered monoids S for which the class of injective S-acts is (P, s)-, (P, a)-, and (P, e)-stable.
Article
Mathematics
Mario Jardon Santos
Summary: The book "Cardinal Invariants on Boolean Algebras" by J. Donald Monk defines and studies many cardinal functions, some of which are generalizations of well-known cardinal characteristics of the continuum. The book also presents a long list of open problems. This study provides definitive answers to several of these problems, focusing on a few cardinal invariants. These problems can be divided into two groups, one concerning the change in cardinal functions when moving from a given infinite Boolean algebra to its simple extensions, and the other comparing a couple of given infinite Boolean algebras and their free product.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Tyler Arant
Summary: This paper discusses the proper approach to making the concept of a Polish space effective. A theorem is proven that demonstrates the absence of a recursive Polish space structure in the effectively open subsets of space X, and strong evidence is explored indicating that the effective structure is instead captured by the effectively open subsets of the product space N x X.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Tapani Hyttinen, Kaisa Kangas
Summary: In this paper, an AEC framework for studying fields with commuting automorphisms is introduced. Fields with commuting automorphisms are connected to difference fields. The paper extends the definition that some authors use for difference rings (or fields) to include several commuting endomorphisms and proves various properties of FCA-classes, including the existence of AP and JEP, the coincidence of Galois types and existential types in existentially closed models, the homogeneity of the class, and a version of type amalgamation theorem. Furthermore, the paper shows that the monster model in this framework is a simple homogeneous structure.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Paul Howard, Eleftherios Tachtsis
Summary: This paper provides a general criterion for Fraenkel-Mostowski models of ZFA and explores six models that satisfy this criterion, which implies LW and DF = F. The paper also examines the choice for well ordered families of well orderable sets in these models. The existence of a model where AC(fin)(WO) is false motivated this study.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Mahya Malekghasemi, Seyed-Mohammad Bagheri
Summary: We prove the Robinson consistency theorem, as well as Craig, Lyndon, and Herbrand interpolation theorems, in linear continuous logic.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Haim Horowitz, Saharon Shelah
Summary: Using a model with a Laver-indestructible supercompact cardinal ?, we establish a model of Z F + DC? in which there are no ?-mad families.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Correction
Mathematics
Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
J. K. Truss
Summary: It is demonstrated that all countable superatomic boolean algebras of finite rank possess the small index property.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Masato Fujita
Summary: The two assertions, one about a definable bijection between bounded and unbounded intervals, and the other about the existence of definable continuous extensions for functions defined on closed subsets, are equivalent for an o-minimal expansion of an ordered group M.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Huishan Wu
Summary: This paper uses techniques of reverse mathematics to study the structure of semisimple rings. A ring is left semisimple if its left regular module is a finite direct sum of simple submodules. The famous Wedderburn-Artin Theorem states that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for this theorem in RCA(0), showing the structure of computable semisimple rings.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Caleb Camrud, Isaac Goldbring, Timothy H. McNicholl
Summary: This article examines the complexity of the quantifier levels of computably presented metric structures in terms of the arithmetical hierarchy. Two types of diagrams, closed and open, are introduced at each level, representing weak and strict inequalities respectively. The closed Sigma(N) and open Sigma(N) diagrams are proven to be Pi(0)(N+1) and Sigma(0)(N) respectively, while the closed Pi(N) and open Pi(N) diagrams are shown to be Pi(0)(N) and Sigma(0)(N+1) respectively. Effective infinitary formulas of continuous logic are introduced and the results are extended to the hyperarithmetical hierarchy. The optimality of the results is demonstrated.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Vincent Guingona, Miriam Parnes
Summary: In this paper, the concept of K-rank is introduced, where K is a strong amalgamation Fraisse class. The K-rank of a partial type is essentially the number of independent copies of K that can be coded within the type. The paper explores K-rank in specific examples such as linear orders, equivalence relations, and graphs, and discusses its relationship with other ranks in model theory, including dp-rank and op-dimension (a notion coined by the authors in previous work).
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Somayyeh Tari
Summary: Assuming M = (M, <, ...) is a weakly o-minimal structure, if there exists an o-minimal structure N such that Def(m)(M) is the collection of all definable subsets of M-m in M for any m in N, then the structure M has the strong cell decomposition property.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Takayuki Kihara, Kenta Sasaki
Summary: Louveau showed that the Gamma-code of a Borel set in a Polish space can be obtained from its Borel code in a hyperarithmetical manner, if it belongs to a Borel Wadge class Gamma. We extend this theorem to Borel functions by proving that the Sigma(t)-code of a Borel function on a Polish space can also be found hyperarithmetically relative to its Borel code. Furthermore, we establish extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Mathematics
Giorgio Laguzzi, Heike Mildenberger, Brendan Stuber-Rousselle
Summary: This paper introduces and investigates versions of Silver and Mathias forcing with respect to lower and upper density. The focus is on properness, Axiom A, chain conditions, preservation of cardinals, and adding Cohen reals. We discover rough forcings that collapse 2ᶿ to ᶿ, while others are surprisingly gentle. We also study the connections between regularity properties induced by these parametrized forcing notions and the Baire property.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Logic
Michael Rescorla
Review of Symbolic Logic
(2023)
Article
Logic
MICHAEL PINSKER, CLEMENS SCHINDLER
JOURNAL OF SYMBOLIC LOGIC
(2023)
Article
Mathematics
Konstantinos A. Beros, Paul B. Larson
Summary: In this paper, we study new examples of ideals with maximal Tukey type and examine the structure of the weak Rudin-Keisler order when restricted to these ideals of maximal Tukey type. We also show the existence of an analytic P-ideal above all other analytic P-ideals in the weak Rudin-Keisler order, mirroring a result on the Tukey order by Fremlin.
ARCHIVE FOR MATHEMATICAL LOGIC
(2023)
Article
Computer Science, Artificial Intelligence
Tikhon Pshenitsyn
Summary: This paper studies categorial grammars based on the Lambek calculus and Lambek-van Benthem calculus LBC. By constructing an LRBVASSAM that generates a non-semilinear set, the conjecture about LBC-grammars generating permutation closures of context-free languages is disproven. The paper also explores the equivalence and properties of LP-grammars and LBC-grammars.
JOURNAL OF LOGIC LANGUAGE AND INFORMATION
(2023)
