Connected components in d-minimal structures

2025/06/27 by Masato Fujita, Fujita, Masato
Computer Science · Decision Sciences · Mathematics · #03C64 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2506.21846

openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ \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.

Citations

• Connectedness in structures on the real numbers: o-minimality and undecidability

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• D-minimal structures
• Structure theorem for i-minimal expansions of the real additive ordered group
• Definable quotients in d-minimal structures
• Groups and rings definable in d-minimal structures
• Definable \mathcal C^r structures on definable topological groups in d-minimal structures