Some Tarskii’s-type problems and logical invariants of algebras
by
Boris Plotkin
Hebrew University

Let Q be a variety of algebras and G be and algebra from Q. Denote by LK_Q(G) the category of elementary sets, i.e., the sets defined by First Order formulas. This category is a logical invariant of the algebra G. We are interested in the situation when the categories LK_Q(G₁) and LK_Q(G₂) are isomorphic. We will define the notion of strong elementary equivalence of algebras. It can be seen that if the algebras G₁ and G₂ are strongly elementary equivalent then they are elementary equivalent. But the converse statement is not true. Besides, strong elementary equivalence implies geometric equivalence of algebras and an isomorphism of the corresponding categories.

Among the numerous arising problems we will distinguish the following Tarskii’s-type problem:

Let F_n and F_m be two noncommutative free groups with n and m generators, respectively. Now it is known that they are elementary equivalent. Is it true that they are strongly elementary equivalent?

This problem is associated with the following question: is it true that every free group is logically noetherian? In other words this is a question about a generalization of Guba’s theorem for free groups.

All questions above relies on a special approach to algebraic logic.

Date received: March 1, 2006