2025/05/28 by Zlil Sela, Sela, Z.
Computer Science · Mathematics · #Algebraic number #Associative property #Closure (psychology) #Diagram #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Logic, programming, and type systems #Rank (graph theory) #Rings and Algebras (math.RA) #Set (abstract data type) #Tuple
paper · pdf · doi:10.48550/arxiv.2505.22755
openalex publication_date 2025/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.