相关论文: On dp-minimal expansions of the integers

We first prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ which is not interdefinable with $\left(\mathbb{Z},+,0,1,<\right)$, then every infinite subset of $\mathbb{Z}$ definable in $\mathcal{Z}$ is…

If $\mathcal{Z}$ is a dp-minimal expansion of a discrete ordered abelian group $(Z,<,+)$ and $\mathcal{Z}$ does not admit a nontrivial definable convex subgroup then $\mathcal{Z}$ is interdefinable with $(Z,<,+)$ and $(Z,<,+)$ is…

We consider the structure $(\mathbb{Z},+,0,|_{p_{1}},\dots,|_{p_{n}})$, where $x|_{p}y$ means $v_{p}(x)\leq v_{p}(y)$ and $v_p$ is the $p$-adic valuation. We prove that its theory has quantifier elimination in the language…

This is a contribution to the classification problem for dp-minimal expansions of $(\mathbb{Z},+)$. Let $S$ be a dense cyclic group order on $(\mathbb{Z},+)$. We use results on "dense pairs" to construct uncountably many dp-minimal…

A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups with only finitely many convex subgroups…

We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or…

Let $\Gamma$ be an infinite discrete subgroup of Gl$_n(\mathbb{C})$. Then either $(\mathbb{R}, <, +, \cdot, \Gamma)$ is interdefinable with $(\mathbb{R}, <, +, \cdot, \lambda^\mathbb{Z})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},…

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many…

We show that the cyclically ordered-abelian groups expanding $(\mathbb{Z};+)$ contain a continuum-size family of dp-minimal structures such that no two members define the same subsets of $\mathbb{Z}$.

Suppose that ${\mathcal M}$ is a model of PA and ${\mathcal N}$ is a countably generated elementary end extension of ${\mathcal M}$. Let ${\mathfrak X}$ be the set of subsets of M that are coded by ${\mathcal N}$. Then ${\mathcal M}$ has a…

This paper investigates expansions of distal structures by a unary subset that arises as the image of a projection map. We first provide a sufficient condition for such an expansion to remain distal. Based on this criterion, we establish…

The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous…

Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathcal{R}$ is generically locally o-minimal. It follows that if…

Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i)…

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do…

The main results of this paper are the construction, both rigourous and intuitive, of "the" intrinsic extension of the set of non negative integers N and the smallest over-field of R set which is continue (according to R.Dedekind). The aim…

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium…

Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval…

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…

Call a (strictly increasing) sequence $(r_{n})$ of natural numbers \emph{regular} if it satisfies the following condition: $r_{n+1}/r_{n}\to\theta\in\mathbb{R}^{>1}\cup\{\infty\}$ and, if $\theta$ is algebraic, then $(r_{n})$ satisfies a…