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Imaginaries, products and the adele ring

Jamshid Derakhshan, Ehud Hrushovski

TL;DR

This work develops a coherent Boolénian-valued framework to understand imaginaries in infinite and reduced powers, reducing imaginaries in a product to those of the factors and enabling weak EI in both full and reduced settings. It then builds a robust theory for atomless and atomized Boolean-valued models, extending to products with a distinguished ideal, and proves weak elimination of imaginaries in these enriched contexts. The adelic case is a central application: leveraging uniform p-adic EI results (HRK) and a Galois-enriched language, the authors obtain a uniform EI basis for the rational adeles described by p-adic imaginaries and lattice-codes, with a blurred version that identifies almost-all primes. The methods combine model-theoretic tools (Feferman–Vaught reductions, liaison groups, stability theory) and descriptive-set-theoretic dichotomies (E_fin, E0, smoothness) to deliver a general approach to imaginaries in restricted powers and in adèlic settings, with potential extensions to algebraic integers and related arithmetic structures.

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.

Imaginaries, products and the adele ring

TL;DR

This work develops a coherent Boolénian-valued framework to understand imaginaries in infinite and reduced powers, reducing imaginaries in a product to those of the factors and enabling weak EI in both full and reduced settings. It then builds a robust theory for atomless and atomized Boolean-valued models, extending to products with a distinguished ideal, and proves weak elimination of imaginaries in these enriched contexts. The adelic case is a central application: leveraging uniform p-adic EI results (HRK) and a Galois-enriched language, the authors obtain a uniform EI basis for the rational adeles described by p-adic imaginaries and lattice-codes, with a blurred version that identifies almost-all primes. The methods combine model-theoretic tools (Feferman–Vaught reductions, liaison groups, stability theory) and descriptive-set-theoretic dichotomies (E_fin, E0, smoothness) to deliver a general approach to imaginaries in restricted powers and in adèlic settings, with potential extensions to algebraic integers and related arithmetic structures.

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 enriched with a predicate for the ideal of finite subsets of . As a special case, using the Hils-Rideau-Kikuchi uniform -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.
Paper Structure (14 sections, 24 theorems, 19 equations)

This paper contains 14 sections, 24 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Products and Boolean-valued models
  3. Imaginaries
  4. Preliminaries
  5. Language of $T^{boole}$
  6. Axioms of $T^{boole}$
  7. Theories of Boolean algebras
  8. Expansions of $T^{boole}$: atomic, atomless and atomic-by-atomless
  9. Imaginaries in infinite products
  10. Definable groups and automorphism groups
  11. Definition of $E^Z$
  12. Imaginaries of atomless Boolean-valued models
  13. Imaginaries with a distinguished ideal.
  14. Adeles

Key Result

Theorem 1

$M^{\sharp}$ admits weak EI to the sorts $D/E^{\sharp}$ (for any $D,E$ as above.) $M^{\flat}$ admits weak EI to the sorts $D/E^{\sharp}$ and $D/E^{\flat}$ (for any $D,E$ as above.)

Theorems & Definitions (61)

  • Theorem
  • Remark 1.3
  • Theorem 2.3
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Example 2.9: Feferman-Vaught Products
  • ...and 51 more