Imaginaries, products and the adele ring

TL;DR

This paper characterizes the imaginary sorts of infinite products and reduced powers, including the ring of rational adeles, using model-theoretic techniques and the Glimm-Efros dichotomy.

Contribution

It extends the understanding of imaginary sorts to infinite products and adeles, combining model theory with descriptive set theory methods.

Findings

Imaginary sorts of infinite products are described in terms of factors.

The imaginary sorts of the ring of rational adeles are characterized.

Methods include the Glimm-Efros dichotomy and model-theoretic orthogonality.

Abstract

We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of finite subsets of $I$. As a special case, using the Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the imaginary sorts of the ring of rational adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning from monadic second order imaginaries to first-order reducts, and for proving a certain ``one-way'' model-theoretic orthogonality within the adelic imaginaries.