Annals of Pure and Applied Logic最新文献

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On classification of continuous first order theories 关于连续一阶理论的分类

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-07-01 Epub Date: 2026-02-02 DOI: 10.1016/j.apal.2026.103729

Karim Khanaki

We present several new characterizations of IP (the independence property) and SOP (the strict order property) for continuous first-order logic and explore their connections to functional analysis and Banach space theory. Furthermore, we propose new dividing lines for unstable theories by examining subclasses of Baire-1 functions. We also explain why one should not expect a perfect analogue of Shelah's theorem—namely, that a theory is unstable if and only if it has IP or SOP—to hold for real-valued logics, particularly in the context of continuous logic.

给出了连续一阶逻辑的独立性和严格序性的几个新特征，并探讨了它们与泛函分析和Banach空间理论的联系。此外，我们通过检验Baire-1函数的子类，提出了不稳定理论的新分界线。我们也解释了为什么我们不应该期待一个完美的类似于Shelah定理——也就是说，一个理论是不稳定的当且仅当它有IP或sop——对于实值逻辑，特别是在连续逻辑的背景下。

引用次数: 0

Positively closed Sh(B)-valued models 正闭Sh(B)值模型

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-06-01 Epub Date: 2026-01-28 DOI: 10.1016/j.apal.2026.103726

Kristóf Kanalas

We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For Set-valued models of coherent theories they coincide.

We prove that if

E = S h (B, τ_{c o h})

for a complete Boolean algebra, then positively closed but not strongly positively closed

E

-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed

E

-valued models.

A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment

L_{κ κ}^{g}

where κ is weakly compact.

我们研究了相干理论的正闭和强正闭拓扑值模型。正闭是一个全局概念（它是根据所有可能的外向同态定义的），而强正闭是一个局部概念（它只涉及模型内的可定义集合）。对于相干理论的集值模型，它们是一致的。我们证明了对于完全布尔代数，如果E=Sh(B,τcoh)，则相干理论的正闭但不是强正闭E值模型存在，然而，存在表征正闭E值模型的另一种局部性质。我们的大部分讨论是在无限量词几何逻辑的背景下给出的，处理片段Lκκg，其中κ是弱紧的。

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引用次数: 0

On algebraic sums, trees and ideals in the Cantor space 关于康托空间中的代数和、树和理想

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-06-01 Epub Date: 2026-01-12 DOI: 10.1016/j.apal.2026.103724

Marcin Michalski, Robert Rałowski, Szymon Żeberski

We work in the Cantor space

2^{ω}

. The results of the paper adhere to the following pattern. Let

I \in {M, N, M \cap N, E}

and T be a perfect, uniformly perfect or Silver tree. Then for every

A \in I

there exists

T^{'} \subseteq T

of the same kind as T such that

A + \underset{n –times}{\underset{︸}{[T^{'}] + [T^{'}] + \dots + [T^{'}]}} \in I

for each

n \in ω

. We also prove weaker statements for splitting trees. For the case

E

we also provide a simple characterization of a basis of

E

. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to

u_{0}

and

v_{0}

, provided that

c

is a regular cardinal.

我们在康托空间2ω中工作。本文的结果遵循以下模式。设I∈{M，N,M∩N,E}， T是一棵完全、一致完全或银树。则对于每一个A∈I，存在与T同类的T’，使得A+[T’]+[T’]+…+[T’]︸n倍∈I，对于每一个n∈ω。我们也证明了劈树的弱命题。对于E，我们也给出了E的基的一个简单表征。我们用这些结果证明了广义Luzin集和广义Sierpiński集的代数和属于u0和v0，只要c是正则基数。

引用次数: 0

Amorphous sets and dual Dedekind finiteness 无定形集与对偶Dedekind有限性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-06-01 Epub Date: 2026-01-08 DOI: 10.1016/j.apal.2026.103723

Yifan Hu , Ruihuan Mao , Guozhen Shen

A set A is dually Dedekind finite if every surjection from A onto A is injective; otherwise, A is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with

ZF

(i.e., Zermelo–Fraenkel set theory without the axiom of choice) that there exists an amorphous set A whose power set

P (A)

is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss (1974) [13]. Nevertheless, we prove in

ZF

that, for all strictly amorphous sets A and all natural numbers n,

P {(A)}^{n}

is dually Dedekind finite, which generalizes a result of Goldstern.

如果从A到A的所有射都是内射，则集合A是对偶Dedekind有限的；否则，A是对偶Dedekind无穷大。无定形集是一种不能被分割成两个无限子集的无限集。严格无定形集是一种无定形集，其中每个划分只有有限多个非单块。证明了存在一个幂集P(A)为对偶Dedekind无穷的无定形集A，与ZF（即不带选择公理的Zermelo-Fraenkel集合理论）是一致的，给出了Truss(1974)[13]问题的一个负解。然而，我们在ZF中证明了，对于所有严格无定形集合A和所有自然数n， P(A)n是对偶Dedekind有限，推广了Goldstern的结果。

引用次数: 0

The computational content of multidimensional discontinuity 多维不连续的计算内容

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-05-01 Epub Date: 2026-01-23 DOI: 10.1016/j.apal.2026.103720

Rupert Hölzl , Keng Meng Ng

The Weihrauch degrees are a tool to gauge the computational difficulty of mathematical problems. Often, what makes these problems hard is their discontinuity. We look at discontinuity in its purest form, that is, at otherwise constant functions that make a single discontinuous step along each dimension of their underlying space. This is an extension of previous work of Kihara, Pauly, Westrick from a single dimension to multiple dimensions. Among other results, we obtain strict hierarchies in the Weihrauch degrees, one of which orders mathematical problems by the richness of the truth-tables determining how discontinuous steps influence the output.

魏氏度是衡量数学问题计算难度的工具。通常，使这些问题变得困难的是它们的不连续性。我们以最纯粹的形式来看待不连续，也就是说，在其他常数函数中，它们沿着其底层空间的每个维度进行单个不连续的步骤。这是Kihara， Pauly， Westrick之前的工作从一个维度到多个维度的延伸。在其他结果中，我们在魏氏度中获得了严格的层次结构，其中一个是通过决定不连续步骤如何影响输出的真值表的丰富程度来排序数学问题。

引用次数: 0

Generic torsion-free groups and Rubin actions 一般无扭群和鲁宾作用

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-05-01 Epub Date: 2025-12-05 DOI: 10.1016/j.apal.2025.103704

Thomas Koberda, Yash Lodha

We use model theoretic forcing to prove that a generic countable torsion-free group does not admit any non-trivial locally moving action on a Hausdorff topological space, and yet admits a rich Rubin poset.

利用模型论强迫证明了一般可数无扭群在Hausdorff拓扑空间上不存在任何非平凡的局部运动，但却存在一个丰富的Rubin偏序集。

引用次数: 0

Groups elementarily equivalent to metabelian Baumslag – Solitar groups and regular bi-interpretability 群基本等价于平衡的Baumslag - Solitar群和正则双可解释性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-05-01 Epub Date: 2025-12-02 DOI: 10.1016/j.apal.2025.103695

Evelina Daniyarova , Alexei Myasnikov

We prove that metabelian Baumslag – Solitar group

BS (1, k)

k > 1

, is (strongly) regularly bi-interpretable with the ring of integers

Z

, and describe in algebraic terms all groups that are elementarily equivalent to

BS (1, k)

证明了metelian Baumslag - Solitar群BS(1,k), k>；1与整数环Z是（强）正则双可解释的，并用代数术语描述了所有与BS（1,k）初等等价的群。

引用次数: 0

Constructive quantifier elimination with a focus on matrix rings 以矩阵环为重点的建设性量词消去

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-05-01 Epub Date: 2025-12-22 DOI: 10.1016/j.apal.2025.103706

Max Illmer, Tim Netzer

We give a sufficient condition for a model theoretic structure B to ‘inherit’ quantifier elimination from another structure A. This yields an alternative proof of one of the main results from [8], namely quantifier elimination for certain matrix rings. The original proof uses model theory, and while it is very elegant and insightful, the proof we propose is much shorter and provides a constructive algorithm.

我们给出了模型理论结构B从另一个结构a“继承”量词消去的一个充分条件。这产生了[8]的一个主要结果的替代证明，即某些矩阵环的量词消去。最初的证明使用了模型理论，虽然它非常优雅和富有洞察力，但我们提出的证明要短得多，并提供了一个建设性的算法。

引用次数: 0

Algebraic independence of the solutions of the classical Lotka-Volterra system 经典Lotka-Volterra系统解的代数独立性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-05-01 Epub Date: 2026-01-02 DOI: 10.1016/j.apal.2025.103707

Yutong Duan, Joel Nagloo

Let

(x_{1}, y_{1}), \dots, (x_{n}, y_{n})

be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system

x^{'} = a x y + b x y^{'} = c x y + d y

where

a, b, c, d \in C ∖ {0}

. We show that if d and b are linearly independent over

Q

, then the solutions are algebraically independent over

C

, that is

t r . d e g_{C} C (x_{1}, y_{1}, \dots, x_{n}, y_{n}) = 2 n

. As a main part of the proof, we show that the set defined by the system in universal differential fields, with d and b linearly independent over

Q

, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general 2d-Lotka-Volterra system.

设（x1,y1），…，（xn,yn）是经典Lotka-Volterra方程组x ' =axy+bxy ' =cxy+dy的不同非常非退化解，其中a，b,c,d∈c∈{0}。我们证明，如果d和b在Q上是线性无关的，那么它们的解在C上是代数无关的，即trd . degcc (x1,y1，…，xn,yn)=2n。作为证明的主要部分，我们证明了在泛微分域中，当d和b在Q上线性无关时，系统所定义的集合是强极小和几何平凡的。我们的技术也允许我们获得一些更一般的2d-Lotka-Volterra系统的部分结果。

引用次数: 0

The category dichotomy for ideals 理想的范畴二分法

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-05-01 Epub Date: 2026-01-07 DOI: 10.1016/j.apal.2025.103717

Alan Dow , Raúl Figueroa-Sierra , Osvaldo Guzmán , Michael Hrušák

We prove that there is an ideal on ω that is not Katětov below nwd and does not have restrictions above

ED

. We also prove that in the Laver model every tall P-ideal is Katětov-Blass above

{ED}_{fin}

and that it is consistent that every Q⁺ ideal is meager.

我们证明了ω上存在一个理想，在nwd以下不为katkattov，在ED以上不受限制。我们还证明了Laver模型中每个高p理想在EDfin以上都是katkattov - blass，并且每个Q+理想都是贫乏的是一致的。

引用次数: 0

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