Problems in algebra inspired by universal algebraic geometry

Abstract

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.