Theory of Interpretations I. Foundations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908660345602048 |
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| author | Daniyarova, Evelina Myasnikov, Alexei |
| author_facet | Daniyarova, Evelina Myasnikov, Alexei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Theory of Interpretations I. Foundations Daniyarova, Evelina Myasnikov, Alexei Logic Group Theory 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. |
| title | Theory of Interpretations I. Foundations |
| topic | Logic Group Theory |
| url | https://arxiv.org/abs/2511.13810 |