Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Theory of Interpretations I. Foundations

Evelina Daniyarova, Alexei Myasnikov

TL;DR

This work establishes a comprehensive framework for interpretability between algebraic structures, distinguishing absolute, regular, invertible, and bi-interpretations and introducing interpretation codes, admissibility conditions, and translation theorems. It analyzes how interpretations act as structured morphisms, enabling systematic transfer of model-theoretic properties and providing uniform strategies to address undecidability and first-order classification via reductions such as Diophantine interpretations. The paper also develops a categorical perspective through homotopy and regular bi-interpretability, and surveys a broad spectrum of applications, including stability, isotypeness, richness, elimination of imaginaries, logical categories, and bi-interpretations with $\mathbb{Z}$ or $\mathbb{N}$, outlining a program for a unified theory of interpretations. The results lay groundwork for future work on regular, strong, and absolute variants, with concrete instances (e.g., matrix groups, Chevalley groups, and rings) illustrating how interpretations can encode deep algebraic and logical phenomena into a transferable framework.

Abstract

This is the first paper in a series in which we lay down the foundations of the theory of interpretations. We systematically study different types of interpretations and their properties. Some of these interpretations are known, while others are new. Each of them serves different purposes. In the last section, we describe applications of interpretations to Diophantine problems, first-order classification, isotypeness, definability of structures by types, elimination of imaginaries, richness, logical categories, and bi-interpretations with Z or N. Additionally, throughout the text, we pose some open questions that naturally arise in this context and provide the most typical examples, usually from algebra. The current literature is plagued by discrepancies and inconsistencies in definitions, concepts, and fundamental applications of interpretations. To address this, we thoroughly examine various principal notions, definitions, and arguments, bringing order to the existing theory. Simultaneously, we develop several key concepts, such as regular interpretations, regular bi-interpretations, and invertible interpretations, and outline their main applications.

Theory of Interpretations I. Foundations

TL;DR

This work establishes a comprehensive framework for interpretability between algebraic structures, distinguishing absolute, regular, invertible, and bi-interpretations and introducing interpretation codes, admissibility conditions, and translation theorems. It analyzes how interpretations act as structured morphisms, enabling systematic transfer of model-theoretic properties and providing uniform strategies to address undecidability and first-order classification via reductions such as Diophantine interpretations. The paper also develops a categorical perspective through homotopy and regular bi-interpretability, and surveys a broad spectrum of applications, including stability, isotypeness, richness, elimination of imaginaries, logical categories, and bi-interpretations with or , outlining a program for a unified theory of interpretations. The results lay groundwork for future work on regular, strong, and absolute variants, with concrete instances (e.g., matrix groups, Chevalley groups, and rings) illustrating how interpretations can encode deep algebraic and logical phenomena into a transferable framework.

Abstract

This is the first paper in a series in which we lay down the foundations of the theory of interpretations. We systematically study different types of interpretations and their properties. Some of these interpretations are known, while others are new. Each of them serves different purposes. In the last section, we describe applications of interpretations to Diophantine problems, first-order classification, isotypeness, definability of structures by types, elimination of imaginaries, richness, logical categories, and bi-interpretations with Z or N. Additionally, throughout the text, we pose some open questions that naturally arise in this context and provide the most typical examples, usually from algebra. The current literature is plagued by discrepancies and inconsistencies in definitions, concepts, and fundamental applications of interpretations. To address this, we thoroughly examine various principal notions, definitions, and arguments, bringing order to the existing theory. Simultaneously, we develop several key concepts, such as regular interpretations, regular bi-interpretations, and invertible interpretations, and outline their main applications.

Paper Structure

This paper contains 29 sections, 70 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Interpretations and their codes
  3. Languages and algebraic structures
  4. Definable sets
  5. Interpretability with parameters
  6. Interpretation codes
  7. Regular interpretability
  8. The admissibility conditions for interpretations
  9. Translation for formulas
  10. Reduction Theorem
  11. Homotopic interpretations
  12. Homotopy with parameters
  13. Regular homotopy
  14. Invertibility and bi-interpretability
  15. Compositions of interpretations
  16. ...and 14 more sections

Key Result

Lemma 1

Suppose that $R$ is a commutative associative unitary ring and $G=\mathrm{UT}_3(R)$. Then there exists a regularization of the interpretation $R\rightsquigarrow \mathrm{UT}_3(R)$ from Example ex:UT2 with descriptor $\phi(y_1,y_2)$, which describes the following statements:

Theorems & Definitions (201)

  • Definition 1
  • Definition 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • ...and 191 more