Uniformly locally o-minimal open core

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

Sedimentary facies of the Tetori Group in the Shiramine, Ishikawa Prefecture

The Mesozoic Tetori Group, exposed in the Shiramine area of Ishikawa Prefecture, is characterized by nonmarine deposits unconformably overlying the Hida Metamorphic Rocks. Sedimentological investigation of the upper part of the Akaiwa Formation (the upper part of the Tetori Group) recognized seven sedimentary facies (channel floor, sand bar, plane beds, crevasse splay, flood plain, vegetated swamp, and soil) and seven cycles; each of them generally appears fining upwards. Stratigraphy of the upper part of the Akaiwa Formation shows a graded change from a gravely fluvial system (the lowest cycle) to a sandy meandering river system (the upper cycles). Nodular calcretes found from the soil facies indicate that the climate was once arid during sedimentation of the upper part of the Akaiwa Formation. However, it is inconsistent with well development of coal seams and carbonaceous mudstone

Expansion of a semi-bounded o-minimal structure by a geometric progression

We demonstrate that an expansion of a semi-bounded o-minimal expansion of the ordered group of reals by an increasing geometric progression is locally o-minimal

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.Comment: arXiv admin note: substantial text overlap with arXiv:2212.0640

Almost o-minimal structures and X\mathfrak X-structures

We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\mathfrak X$-sets and $\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\mathfrak X$-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded $\mathfrak X$-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups $\mathcal M=(M,<,0,+,\ldots)$. Let $\{A_\lambda\}_{\lambda\in\Lambda}$ be a finite family of definable subsets of $M^{m+n}$. Take an arbitrary positive element $R \in M$ and set $B=]-R,R[^n$. Then, there exists a finite partition into definable sets \begin{equation*} M^m \times B = X_1 \cup \ldots \cup X_k \end{equation*} 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 either $X_i \cap A_\lambda = \emptyset$ or $X_i \subseteq A_\lambda$ for any $1 \leq i \leq k$ and $\lambda \in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b \in M^m$. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.Comment: arXiv admin note: text overlap with arXiv:1912.0578

Notes on definably complete locally o-minimal expansions of ordered groups

We study definably complete locally o-minimal expansions of ordered groups in this paper. A definable continuous function defined on a closed, bounded and definable set behave like a continuous function on a compact set. We demonstrate uniform continuity of a definable continuous function on a closed, bounded and definable set and Arzela-Ascoli-type theorem. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.Comment: arXiv admin note: text overlap with arXiv:2010.0242