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Masato Fujita

Publications and source records attributed to Masato Fujita.

At least 19 recordsLinked to original sources
arXiv2026

A note on generic $n$-partite graphs

An $n$-partite graph is a graph such that every vertex has a color in $\{1,\ldots,n\}$ and every two vertices of the same color are not adjacent. We study the model comparisons of the theories of $n$-partite graph and $K_{\overline{m}}$-free $n$-partite graph, where $K_{\overline{m}}$ is a complete graph of a given size. The model companion of the theory of $n$-partite graph is simple and has IP. The model companion of the theory of $K_{\overline{m}}$-free $n$-partite graph has $\operatorname{TP}_2$, $\operatorname{SOP}_3$ and $\operatorname{NSOP}_4$ if $n > 2$. Forking independence coincides with dividing independence in this theory.

math.LO↗
arXiv2025

Cardinality of the sets of dimension functions in ordered structures

We compute the cardinality $\mathfrak n_{\dim}(\mathcal M)$ of the sets of dimension functions on the ordered structures $\mathcal M$. The inequality $\mathfrak n_{\dim}(\mathcal M) \leq 1$ holds if $\mathcal M$ is a d-minimal expansion of an ordered group. If $\mathcal M$ is o-minimal and $\mathfrak n_{\dim}(\mathcal M)<\infty$, there exists a positive integer $m$ such that $\mathfrak n_{\dim}(\mathcal M)=2^m-1$. For every positive integer $m$, there exists a weakly o-minimal expansion $\mathcal M$ of an ordered divisible Abelian group such that $\mathfrak n_{\dim}(\mathcal M)=m$.

math.LO↗
arXiv2025

Connected components in d-minimal structures

For a given d-minimal expansion $\mathfrak R$ of the ordered real field, we consider the expansion $\mathfrak R^\natural$ of $\mathfrak R$ generated by the sets of the form $\bigcup_{S \in \mathcal C}S$, where $\mathcal C$ is a subfamily of the collection of connected components of an $\mathfrak R$-definable set. We prove that $\mathfrak R^{\natural}$ is d-minimal. A similar assertion holds for almost o-minimal expansions of ordered groups.

math.LO↗
arXiv2025

Nonvaluational ordered Abelian groups of finite burden

Consider an expansion $\mathcal R=(R,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X \subseteq U$. We further assume that $\mathcal R$ is nonvaluational, that is, for every nonempty definable subsets $A,B$ of $R$ with $A <B$ and $A \cup B=R$, $\inf\{b-a\;|\;a \in A, b \in B\}=0$. Then, $\mathcal R$ is $*$-locally weakly o-minimal. We also give a complete description of sets definable in a definably complete expansion of ordered group of burden two if it defines an infinite discrete set.

math.LO↗
arXiv2025

Morse functions definable in d-minimal structures

Fix a d-minimal expansion of an ordered field. We consider the space $\mathcal D^p(M)$ of definable $\mathcal C^p$ functions defined on a definable $\mathcal C^p$ submanifold $M$ equipped with definable $\mathcal C^p$ topology. The set of definable $\mathcal C^p$ Morse functions is dense in $\mathcal D^p(M)$.

math.LO↗
arXiv2024

Morse theory in definably complete d-minimal structures

Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 \le r <\infty$. We prove that the set of definable Morse functions is open and dense in the set of definable $C^r$ functions on $X$ with respect to the definable $C^2$ topology.

math.LO↗
arXiv2024

There exists a d-minimal expansion of the $\mathbb R$-vector space over $\mathbb R$ which defines every sequence

There exists a d-minimal expansion of the $\mathbb R$-vector space over $\mathbb R$ which defines every sequence. In this paper, we prove this assertion and the following more general assertion: Let $\mathcal R$ be either the ordered $\mathbb R$-vector space structure over $\mathbb R$ or the ordered group of reals. A first-order expansion of $\mathcal R$ by a countable subset $D$ of $\mathbb R$ and a compact subset $E$ of $\mathbb R$ of finite Cantor-Bendixson rank is d-minimal if $(\mathcal R,D)$ is locally o-minimal.

math.LO↗
arXiv2024

Uniform local weak o-minimality and $*$-local weak o-minimality

We propose the notions of uniform local weak o-minimality and $*$-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o-minimal structures of the second kind and for locally o-minimal structures under the assumption called the univariate $*$-continuity property. We also prove that several formulas for dimension of definable sets which hold in definably complete locally o-minimal structures also hold in $*$-locally weakly o-minimal structures possessing the univariate $*$-continuity property.

math.LO↗
arXiv2024

Michael's selection theorem in general d-minimal structures

Thamrongthanyalak demonstrated a definable version of Michael's selection theorem in d-minimal expansions of the real field. We generalize this result to the case in which the structures are d-minimal expansions of ordered fields $\mathcal F=(F,<,+,\cdot,0,1,\ldots)$. We also show that we can choose a definable continuous selection $f$ of a lower semi-continuous map $T:E \rightrightarrows F$ so that $f(x)$ is contained in the interior of $T(x)$ when the interior is not empty.

math.LO↗
arXiv2023

Definable quotients in d-minimal structures

We consider d-minimal expansions of ordered fields. We demonstrate the existence of definable quotients of definable sets by definable equivalence relations when several technical conditions are satisfied. These conditions are satisfied when there is a definable proper action of a definable group $G$ on a locally closed definable subset $X$ of $F^n$, where $F$ is the universe.

math.LO↗
arXiv2023

Quasi-quadratic modules in pseudo-valuation domain

We study quasi-quadratic modules in a pseudo-valuation domain $A$ whose strict units admit a square root. Let $\mathfrak X_R^N$ denote the set of quasi-quadratic modules in an $R$-module $N$, where $R$ is a commutative ring. It is known that there exists a unique overring $B$ of $A$ such that $B$ is a valuation ring with the valuation group $(G,\leq)$ and the maximal ideal of $B$ coincides with that of $A$. Let $F$ be the residue field of $B$. In the above setting, we found a one-to-one correspondence between $\mathfrak X_A^A$ and a subset of $\prod_{g \in G,g \geq e} \mathfrak X_{F_0}^F$.

math.AC↗
arXiv2023

Quasi-quadratic modules in valuation ring and valued field

This is a revised version of the previous version with a new appendix consisting of characteristic two case. We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main theorem is a structure theorem of quasi-quadratic modules in a subring $A$ of a $2$-henselian valued field $(K,{\bf val})$ whose residue class field $F$ of characteristic $\neq 2$. We further assume that the valuation ring $B$ is contained in $A$. Set $H={\bf val}(A^\times)$ and $G_{\geq e}=\{g \in G\;|\; g \geq e\}$. The notation $\mathfrak X_R$ denotes the set of all the quasi-quadratic modules in a commutative ring $R$. Our structure theorem asserts that there exists a one-to-one correspondence between $\mathfrak X_A$ and a subset $\mathcal T_F^{ H \cup G_{\geq e}}$ of $\prod_{g \in H \cup G_{\geq e}}\mathfrak X_F$. We explicitly construct the map $Θ: \mathfrak X_A \rightarrow \mathcal T_F^{ H \cup G_{\geq e}}$ and its inverse. We also give explicit expressions of $Θ(\mathcal M \cap \mathcal N)$ and $Θ(\mathcal M+\mathcal N)$ for $\mathcal M, \mathcal N \in \mathfrak X_A$. In addition, we briefly investigate the case in which the field $F$ is of characteristic two in the appendix as well.

math.AC↗
arXiv2023

Definable compactness in definably complete locally o-minimal structures

We demonstrate that And\'ujar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures. As an application, we show that a definably simple definable topological group which is regular, Hausdorff and definably compact as a definable topological space is either discrete or definably connected. We also study the definable quotient of definable continuous actions by definably compact definable topological groups.

math.LO↗