Abstract. Some basic notions of classical algebraic geometry can be defined in arbitrary varieties of algebras Θ. For every algebra H in Θ one can consider algebraic geometry in Θ over H. Correspondingly, algebras in Θ are considered with the emphasis on equations and geometry. We give examples of geometric properties of algebras in Θ and of geometric relations between them. The main problem considered in the paper is when different H1 and H2 have the same geometry.

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 Θ 0 be a category of finitly generated free algebras in the variety of algebras Θ. Solutions to problems in algebraic geometry over Θ are often determined by the structure of the group of automorphisms AutΘ 0 of category Θ 0. Here we consider two varieties Θ: noetherian modules and Lie algebras. We show that every automorphism in AutΘ 0, where Θ is the variety of modules over noetherian rings, is semi-inner. A similar result for the variety of Lie algebras over a infinite field has been recently obtained in [9]. Here we present a different approach allowing us to shorten the proof in the Lie case.

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.

in algebra inspired by universal algebraic geometry

equivalence, geometrical similarity

We prove that every automorphism of the category of free Lie algebras is a semi-inner automorphism. This solves the problem 3.9 from [19] for Lie algebras.

Abstract. In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ 0) of automorphisms of the category Θ 0 of finitely generated free algebras of Θ is of great importance. In this paper, we prove that all automorphisms of categories of free S-acts are semi-inner, which solves a variation of Problem 12 in [12] for monoids. We also give a description of automorphisms of categories of finitely generated free algebras of varieties of unary algebras, and show that among varieties of unary algebras only the variety of mono-unary algebras is perfect [7]. 1.