Automorphic equivalence of one-sorted algebras
  One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? There are various interpretations of the sentence "Two algebras have the same algebraic geometry". One of these is automorphic equivalence of algebras, which is discussed in this paper, and the other interpretation is geometric equivalence of algebras. In this paper we consider very wide and natural class of algebras: one sorted algebras from IBN variety. The variety is called an IBM variety if two free algebras W(X), W(Y) of this variety are isomorphic if and only if the powers of sets X and Y coincide. In the researching of the automorphic equivalence of algebras we must study the group of automorphisms of the category of the all finitely generated free algebras of our variety and the group of its automorphisms. By [PZ, Theorem 2], if we deal with an IBN variety of one-sorted algebras, then every strongly stable (see Definition 3.1) automorphism of our category defines the other algebraic structure on every algebra H of our variety, such that this algebra with the new algebraic structure automorphically equivalent to the algebra H (Theorem 4.1), i.e., has the same algebraic geometry. From this we conclude the necessary and sufficient conditions for two algebras to be automorphically equivalent. We formulate these conditions by using the notion of geometric equivalence of algebras. It means that we reduce automorphic equivalence of algebras to the simpler notion of geometric equivalence. This paper is a continuation of the research which was started in [PZ]. 
A.Tsurkov