Some supplements to Feferman-Vaught related to the model theory of adeles

TL;DR

This paper advances the model theory of finite adeles over number fields by analyzing their structure in ring language, including valuations and hyperfields, providing foundational insights and systematic treatments.

Contribution

It introduces a ring-theoretic approach to the model theory of adeles, including valuation and hyperfield structures, extending prior work with new foundational results.

Findings

Systematic treatment of product valuation and valuation monoid.

Connections between adelic hyperfields and Basarab-Kuhlmann formalism.

Foundational results for model theory of finite adeles.

Abstract

We give foundational results for the model theory of the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab-Kuhlmann formalism.