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

TL;DR

This paper proves the existence of a d-minimal expansion of the real vector space that can define every sequence, extending the understanding of minimality in ordered structures.

Contribution

It establishes the existence of a d-minimal expansion of the real vector space over reals that can define all sequences, generalizing previous results.

Findings

Existence of a d-minimal expansion defining all sequences

Extension of d-minimality to structures with specific subsets

Generalization to ordered real vector spaces and groups

Abstract

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.