Theory of Interpretations I. Foundations

TL;DR

This foundational paper systematically develops the theory of interpretations, clarifying existing concepts, introducing new notions, and exploring applications in algebra, logic, and Diophantine problems.

Contribution

It establishes a rigorous framework for interpretations, introduces concepts like regular and invertible interpretations, and addresses inconsistencies in the existing literature.

Findings

Clarified and unified definitions of interpretations.

Introduced new concepts such as regular and invertible interpretations.

Applied interpretations to various problems in algebra and logic.

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.