Model Theory of Local Real Closed SV-Rings of Finite Rank

TL;DR

This paper develops a model-theoretic framework for local real closed SV-rings of finite rank, providing structure theorems, elementary classes, and quantifier elimination results for these rings.

Contribution

It introduces a complete, decidable, and NIP model theory for local real closed SV-rings of finite rank, including their model completion and quantifier elimination.

Findings

Models are n-fold fiber products of real closed valuation rings.

Theories T_{n,1} and T_{n,2} are complete, decidable, and NIP.

Model completion and quantifier elimination are established for these classes.

Abstract

This note begins the model-theoretic study of local real closed SV-rings of finite rank; to this end, a structure theorem for reduced local SV-rings of finite rank is given and branching ideals in local real closed rings of finite rank are analysed. The class of local real closed SV-rings of rank $ n \in \mathbb{N}^{\geq 2} $ is elementary in the language of rings $ \mathscr{L} := \{ +, -, \cdot, 0, 1 \} $ and its $ \mathscr{L} $-theory $ T_n $ has a model companion $ T_{n,1} $; models of $ T_{n,1} $ are $ n $-fold fibre products $ (((V_1 \times_{\textbf{k}} V_2) \times_{\textbf{k}}V_3) \dots \times_{\textbf{k}} V_{n-1})\times_{\textbf{k}} V_{n} $ of non-trivial real closed valuation rings $ V_i $ with isomorphic residue field $\mathbf{k}$. The $ \mathscr{L} $-theory $ T_{n,1} $ is complete, decidable, and NIP. After enriching $ \mathscr{L} $ with a predicate for the maximal ideal, models of $ T_n $ have prime extensions in models of $ T_{n,1} $, and $ T_{n,1} $ is the model completion of $ T_n $ in this enriched language. A quantifier elimination result for $ T_{n,1} $ is also given. The class of those local real closed SV-rings of rank $ n\in \mathbb{N}^{\geq 2} $ which are $ n $-fold fibre products $ (((V_1 \times_{W} V_2) \times_{W}V_3) \dots \times_{W} V_{n-1})\times_{W} V_{n} $ of non-trivial real closed valuation rings $ V_i $ along surjective morphisms $ V_i \twoheadrightarrow W $ onto a non-trivial domain $ W $ is elementary in the language of rings, and its $ \mathscr{L} $-theory $ T_{n,2} $ is also complete, decidable, and NIP; after enriching $ \mathscr{L} $ with predicates for the maximal ideal and the unique branching ideal, $ T_{n,2} $ is model complete.