Abstract. Let V be a variety of universal algebras. A method is suggested for describing automorphisms of a category of free V-algebras. Applying this general method all automorphisms of such categories are found in two cases: 1) V is the variety of all free associative K−algebras over an infinite field K and 2) V is the variety of all representations of groups in unital R−modules over a commutative associative ring with unit. It turns out that they are almost inner in a sense.

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms. 1

Abstract. Given a variety V of universal algebras. A new approach is suggested to characterize algebraically automorphisms of the category of free V-algebras. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category are inner or not. This question is important for universal algebraic geometry [5, 9]. Most of results will actually be proved to hold for arbitrary categories with a represented forgetful functor.

Abstract. Let A = A(x1,...,xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x1,...,xn}, End A be the semigroup of endomorphisms of A, and Aut EndA be the group of automorphisms of the semigroup EndA. We investigate the structure of the groups Aut EndA and Aut A ◦ , where A ◦ is the category of finitely generated free algebras from A. We prove that the group Aut EndA is generated by semiinner and mirror automorphisms of EndF and the group Aut A ◦ is generated by semi-inner and mirror automorphisms of the category A ◦. This result solves an open Problem formulated in [14]. 1.

Abstract. We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F(x1,...,xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1,...,xn}, EndF be the semigroup of endomorphisms of F, and Aut EndF be the group of automorphisms of the semigroup EndF. We investigate structure of the group Aut End F and its relation to the algebraical and categorical equivalence of algebras from Θ. We define a wide class of R1MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Φ of semigroup EndF, where F is a free finitely generated Lie algebra over an R1MF-domain, is semi-inner. This solves the Problem 5.1 left open in [21]. As a corollary, semi-innerity of all automorphism of the category of free Lie algebras over R1MF-domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R1MFdomains are clarified. The group Aut EndF for the variety of m-nilpotent associative algebras over R1MF-domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R1MF-domains is obtained. We give an example of the variety Θ of linear algebras over a Dedekind domain such that not all automorphisms of Aut EndF are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields. 1.

Prof. Boris I. Plotkin [1, 2] drew attention to the question when an equivalence between two categories is isomorphic as a functor to an isomorphism between them. It turns out that it is quite important for universal algebraical geometry and concerns mainly the categories Θ 0 (X) of free universal algebras of some variety Θ free generated by finite subsets of X. In the paper, a complete answer to the Plotkin’s question is given: there are no proper autoequivalences of the category Θ 0 (X). Also some connected problems are discussed.

and Related Areas, sponsored by the Minerva Foundation (Germany)

in algebra inspired by universal algebraic geometry

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 automorphisms 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) isomorphism

This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [Pl],[Ts]) we must consider the variety Θ, to which our algebras belongs, the category Θ 0 of all finitely generated free algebras of Θ and research how the group AutΘ 0 of all the automorphisms of the category Θ 0 are different from the group InnΘ 0 of the all inner automorphisms of the category Θ 0. An automorphism Υ of the arbitrary category K is called inner, if it is isomorphic as functor to the identity automorphism of the category K, or, in details, for every A ∈ ObK there exists s Υ A: A → Υ (A) isomorphism