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.

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.

In this paper is proved that the group of automorphisms of semigroup End(P[X]), if P is algebraically closed field, is generated by semi-inner automorphisms. 1

Abstract. Last years a number of papers were devoted to describing automorphisms of semigroups of endomorphisms of free finitely generated universal algebras of some varieties: groups, semigroups, associative commutative algebras, inverse semigroups, modules, Lie algebras and some others (see references). All these researches were inspired by the questions prof. B. Plotkin set in connection with so called universal algebraic geometry [11], [12]. The aim of this paper is to suggest a method of describing Aut End(F) for a free algebra F of an arbitrary variety of universal algebras. This method allows to obtain easily all known results as well as new ones. 1. introduction The aim of this paper is to suggest a method of describing Aut End(F) where F is a free algebra of an arbitrary variety of universal algebras. The interest to this problem was inspired in most cases by questions which were set by B. Plotkin in connection with universal algebraic geometry developed in his papers (see for example [11], [12]).

Abstract. We describe the automorphism group of the endomorphism semigroup End(K[x1,..., xn]) of ring K[x1,..., xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin ( [18], Problems 12 and 15). More precisely, we prove that if ϕ ∈ AutEnd(K[x1,..., xn]) then there exists a semi-linear automorphism s: K[x1,..., xn] → K[x1,..., xn] such that ϕ(g) = s ◦g ◦s −1 for any g ∈ End(K[x1,..., xn]). This extends the result by A. Berzins obtained for an infinite field K. 1.