Related papers: Tame structures and open cores
We demonstrate that the open core of a definably complete expansion of a densely linearly ordered abelian group is locally o-minimal if and only if any definable closed subset of $R$ is either discrete or contains a nonempty open interval.…
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One…
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…
We introduce and study some general principles and hierarchical properties of expansions and restrictions of structures and their theories The general approach is applied to describe these properties for classes of $\omega$-categorical…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality. In particular, we prove that we can endow a definable group with a definable topology making…
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…
Every bounded definable open set is a union of finitely many open strong cells in a weakly o-minimal expansion of a real closed field. We prove this fact and another theorem similar to it.
In this paper, we introduce and study the concepts of semi open SOM) and semi closed (SCM) M-sets in multiset topological spaces.With this generalization of the notions of open and closed sets in M-topology, we generalize the concept of…
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of…
We give a survey on recent developments in the model theory of valued fields since the introduction of the notion of ``tame valued field'', and of the modifications and generalizations of this notion.
The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches…
Let $f$ be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous…
This paper studies the structure of core sets under different similarity classes. We investigate the influence of factors of the minimal polynomial with different degrees on the structure of core sets. When $F$ is a finite field of prime…
Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial…
Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in…
We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the…
We present a systematic study of join-extensions and join-completions of ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from…
We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed…
We show some basic facts about dp-minimal ordered structures. The main results are : dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure…