  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].
