On VC-minimal fields and dp-smallness
Vincent Guingona
TL;DR
This paper proves that VC-minimal ordered fields are real closed and introduces the concept of dp-smallness, which helps characterize algebraic theories like ordered groups and fields.
Contribution
It introduces the notion of dp-smallness, a new property between convex orderability and dp-minimality, and applies it to characterize algebraic structures.
Findings
VC-minimal ordered fields are real closed
Dp-small ordered groups are abelian divisible
Dp-small ordered fields are real closed
Abstract
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Rings, Modules, and Algebras · Computability, Logic, AI Algorithms