Connected components in d-minimal structures

TL;DR

The paper proves that expanding a d-minimal structure by unions of connected components preserves d-minimality, extending the understanding of definable sets in ordered structures.

Contribution

It introduces a new expansion of d-minimal structures by connected components and proves this expansion remains d-minimal, broadening the class of definable sets.

Findings

The expansion $rak R^ atural$ is d-minimal.

A similar result holds for almost o-minimal expansions.

Connected component unions preserve minimality properties.

Abstract

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.