ar X iv : m at h / 05 09 03 2 v 3 [ m at h . G M ] 1 8 Fe b 20 06 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 Θ is called an IBM variety if two free algebras W (X) , W (Y) ∈ Θ 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 automor-phisms of the category Θ 0 of the all finitely generated free algebras of Θ and the group of its automorphisms AutΘ 0. An automorphism Υ of the category K is called inner if it is isomorphic to the identity automorphism or, in other words, if for every A ∈ ObK there exists s Υ A : A → Υ (A) iso-morphism of these objects of the category K and for every α ∈ Mor K (A, B) the diagram A − → s Υ A Υ (A) ↓ α Υ (α) ↓ B s Υ B − → Υ (B) is commutative. By [PZ, Theorem 2], if Θ is an IBN variety of one-sorted algebras, then every automorphism Ψ ∈ AutΘ 0 can be decomposed: Ψ = ΥΦ, where Υ, Φ ∈ AutΘ 0 , Υ is an inner automorphism of and Φ 1 is a strongly stable one (see Definition 3.1). In this situation every strongly stable automorphism defines the other algebraic structure on every algebra H ∈ Θ , such that the algebra H * with this structure also belongs to our variety Θ and (Theorem 4.1) even automorphically equivalent to the algebra H, 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].