A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909626480459776 |
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| author | Sela, Z. |
| author_facet | Sela, Z. |
| contents | The JSJ decomposition and the Makanin-Razborov diagram were proved to be essential in studying varieties over free groups, semigroups and associative algebras. In this paper we suggest a unified conceptual approach to the applicability of these structures over all these algebraic categories. With a variety over each of these algebraic categories we naturally associate a set of tuples in a free group. Then we show how to associate a Makanin-Razborov diagram with any set of tuples over a free group. Furthermore, in case the MR diagram that is associated with a set of tuples is single ended, we prove that there is a canonical Makanin-Razborov diagram that can be associated with such a set. This canonical diagram is a main key in studying varieties over free semigroups, associative algebras and Lie algebras, and encodes the global structure of these varieties. It enables us to define a (pseudo) closure of a set of tuples over each of the algebraic objects, associate a rank with it (analogous to Shelah and Lascar ranks), and over free groups the closure provides a canonical envelope that is essential in studying the structure and the properties of definable sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_22755 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I Sela, Z. Group Theory Logic Rings and Algebras The JSJ decomposition and the Makanin-Razborov diagram were proved to be essential in studying varieties over free groups, semigroups and associative algebras. In this paper we suggest a unified conceptual approach to the applicability of these structures over all these algebraic categories. With a variety over each of these algebraic categories we naturally associate a set of tuples in a free group. Then we show how to associate a Makanin-Razborov diagram with any set of tuples over a free group. Furthermore, in case the MR diagram that is associated with a set of tuples is single ended, we prove that there is a canonical Makanin-Razborov diagram that can be associated with such a set. This canonical diagram is a main key in studying varieties over free semigroups, associative algebras and Lie algebras, and encodes the global structure of these varieties. It enables us to define a (pseudo) closure of a set of tuples over each of the algebraic objects, associate a rank with it (analogous to Shelah and Lascar ranks), and over free groups the closure provides a canonical envelope that is essential in studying the structure and the properties of definable sets. |
| title | A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I |
| topic | Group Theory Logic Rings and Algebras |
| url | https://arxiv.org/abs/2505.22755 |