A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I

Abstract: 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.