Masato Fujita
This paper discusses sufficient conditions for a definably complete densely linearly ordered expansion of an abelian group having the uniformly locally o-minimal open cores of the first/second kind and strongly locally o-minimal open core, respectively.
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In this paper using Sperner's lemma for modified partition of a simplex we will constructively prove Brouwer's fixed point theorem for sequentially locally non-constant and uniformly sequentially continuous functions.
We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind having definable bounded multiplication being affine.
Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where $X$ is a definable set and $R$ is the universe of the structure. We demonstrate the inequality $\dim(f(X)) \leq \dim(X)$ in this paper. As a corollary, we get that the set of the points at which $f$ is discontinuous is of dimension smaller than $\dim(X)$. We also show that the structure is defiably Baire in the course of the proof of the inequality.
We investigate continuous functions definable in a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group (DCULOAS structure). We prove a variant of the Arzela-Ascoli theorem for uniformly continuous definable functions and the following assertion: Consider the parameterized function $f:C \times P \rightarrow M$ which is equi-continuous with respect to $P$. The projection image of the set at which $f$ is discontinuous to the parameter space $P$ is of dimension smaller than $\dim P$ when $C$ is closed and bounded. In addition, we …
We prove a structure theorem for periodic locally soluble groups satisfying a chain condition on intersections of relatively uniformly definable subgroups using results from the theory of stable groups. The result in particular shows that these groups are soluble, thus giving a model-theoretic and much simpler proof of a special case of a theorem of Bryant and Hartley on the solubility of periodic locally soluble groups satisfying the minimal condition on centralizers.
Uniform Local Tabularity in Intuitionistic Logic
本文研究直觉主义逻辑中一致局部表格性的强化概念，通过Heyting代数的迭代蕴涵生成，证明n一致局部有限Heyting代数构成簇，并给出n<3时的公理化，同时构造反例回答Shehtman问题。
We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions implies Sperner's lemma for a simplex. Since the existence of approximate fixed points is derived from Sperner's lemma, our Brouwer's fixed point theorem is equivalent to Sperner's lemma.
Invariance of domain in locally o-minimal structures
本文研究了局部o-极小结构中定义连续单射映射的开放性性质，证明了其在相同维度欧几里得空间中的映射特性。
There exist NIP and non-NTP$_2$ theories satisfying all the following conditions: It is not o-minimal; All models are strongly locally o-minimal; It has a model which is an expansion of the linearly ordered abelian group over the reals $(\mathbb R\;;\;0,+,<)$. We construct these examples using the notion of simple product.
Uniform local weak o-minimality and $*$-local weak o-minimality
本文引入一致局部弱 o-极小性与 $*$-局部弱 o-极小性，并证明相关结构在一元 $*$-连续性性质下的局部单调性定理及可定义集维数公式。
We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.
Locally o-minimal open core
该研究证明了在可定义完备扩张中，开核为局部 o-极小的条件是所有可定义闭子集要么离散要么包含非空开区间。
In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allows us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable cove…
Definable quotients in locally o-minimal structures
本文研究了局部o-最小结构中可定义集的可定义商的存在性，通过局部闭子集和可定义群作用的条件来保证结果。
Let $M$ be a $C$-minimal structure and $T$ its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than $\infty$ ordered by inclusion). We present a description of definable locally constant functions $f:M\rightarrow T$ in $C$-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a description of definable subsets of $T$ in one variable and analogues of known results in algebraically closed valued fields.
Definable compactness in definably complete locally o-minimal structures
本文将 o-极小结构中可定义紧性的经典结果推广到可定义完全局部 o-极小结构，并应用于刻画可定义简单拓扑群的离散性或连通性，同时研究了可定义紧群作用下的可定义商。
We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact. A definable set $X$ is of dimension equal to the rank of $X$ over the set of parameters of a formula defining the set $X$. The structure $\mathcal M$ is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure $\mathcal M$ also coincides …
In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.
We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, in the latter part of this paper, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fiel…
Notes on definably complete locally o-minimal expansions of ordered groups
本文研究有序群的可定义完备局部 o-minimal 扩张，证明闭有界可定义集上连续函数的一致连续性与 Arzela-Ascoli 型定理，并建立带管状邻域的特殊子流形有限分解。
Non-elementary categoricity and projective locally o-minimal classes
本文将光滑复代数簇族的覆盖与实数 o-极小扩张中局部可定义的结构类 $\mathcal{U}$ 相关联，证明了其子模型上的 $\aleph_0$-齐次性、稳定性及 $\aleph_1$-范畴性，并在一维情形下修改得到所有不可数基数下范畴的抽象初等类。
We demonstrate the following uniform local definable cell decomposition theorem in this paper. Consider a structure $\mathcal M = (M, <,0,+, \ldots)$ elementarily equivalent to a locally o-minimal expansion of the group of reals $(\mathbb R, <,0,+)$. Let $\{A_λ\}_{λ\inΛ}$ be a finite family of definable subsets of $M^{m+n}$. There exist an open box $B$ in $M^n$ containing the origin and a finite partition of definable sets $M^m \times B = X_1 \cup \ldots \cup X_k$ such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and $X_i \cap A_λ= \emp…
We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally definable in a fixed o-minimal and polynomially bounded reduct. As an application we show that in certain o-minimal structures definable functions are piecewise implicitly defined over the basic functions in the language.
Approximation and zero set of definable functions in a definably complete locally o-minimal structure
本文研究了局部o-最小结构中可定义函数的近似与零集，提出了可定义流形的嵌入定理及微分同胚性结论。
Let $G$ be a locally compact Polish group. A metrizable $G$-flow $Y$ is called model-universal if by considering the various invariant probability measures on $Y$, we can recover every free action of $G$ on a standard Lebesgue space up to isomorphism. Weiss has shown that for countable $G$, there exists a minimal, model-universal flow. In this paper, we extend this result to all locally compact Polish groups.
It is well known that strongly minimal groups are commutative. Whether this is true for various generalisations of strong minimality has been asked in several different settings (see Hyttinen [2002], Maesono [2007], Pillay and Tanović [2011]). In this note we show that the answer is positive for groups with locally modular homogeneous pregeometries.
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