Isotypical Equivalence of Rings and Groups

In memory of Zhenya Plotkin

Two groups are said to be isotypically equivalent (or simply isotypic) if they realize exactly the same types. Recall that the type of a tuple of elements (a_1, …, a_m) in a group G is the set of all formulas F(x_1, \dots, x_n)  in the language of group theory that hold for the tuple (a_1, … , a_m) in G. In this situation, we say that this type is realized in G. This talk will focus on a problem posed by Boris Plotkin: Are any two finitely generated isotypic groups necessarily isomorphic? Zhenya Plotkin was deeply intrigued by this question, as it can be answered positively for certain specific classes of groups. However, at present, there are no general methods in model theory or algebra that allow us to tackle this problem in full generality. In fact, this question is also very interesting for other structures, such as rings or semigroups. I will touch on this in my talk as well.