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 Θ be an arbitrary variety of algebras and let Θ 0 be the category of all free finitely generated algebras from Θ. We study automorphisms of such categories for special Θ. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are also considered. This topic relates to algebraic geometry in arbitrary variety of algebras Θ. 1. Motivations 1.1. The main problem and automorphisms of free objects. We consider an arbitrary variety of algebras Θ. For any Θ denote by Θ0 the category of all free in Θ algebras W = W (X), where X is finite. In order to avoid set-theoretic problems we view all X as subsets of a universal infinite set X0. Our main goal is to study automorphisms of the category Θ0 and the corresponding group Aut Θ0. The study of automorphisms of the category Θ0 is tied to the study of automorphisms

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

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Assume that Θ is an arbitrary variety of groups. Let W(X) be a free group of the variety Θ over the finite set X and G is a group in this variety (G ∈ Θ). We can consider the ”affine space over the group G”: HomΘ(W(X), G). For every set T ⊂ W(X) we can consider the ”algebraic variety”

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

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

In the paper we prove (Theorem 8.1) that there exists a continuum of non isomorphic simple modules over KF2, where F2 is a free group with 2 generators (compare with [Ca] where a continuum of non isomorphic simple 2-generated groups is constructed). Using this fact we give an example of a non action type logically Noetherian representation (Section 9). In general, the topic of this paper is the action type algebraic geometry of representations of groups. For every variety of algebras Θ 1 and every algebra H ∈ Θ we can consider an algebraic geometry in Θ over H. Algebras in Θ may be many sorted (not necessarily one sorted) algebras. A set of sorts Γ is fixed for each Θ. This theory can be applied to the variety of representations of groups over fixed commutative ring K with unit. We consider a representation as two sorted algebra (V,G), where V is a K-module, and G is a group acting on

The problem of the classification of the nilpotent class 2 torsion free groups up to the geometrically equivalence.

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.