Let V be a variety of universal algebras. We suggest a method for describing automorphisms of the category of free V-algebras. All automorphisms of such categories are found in two cases: (1) V is the variety of all associative K-algebras over an infinite field K; (2) V is the variety of all representations of groups in unital R-modules over a commutative associative ring R with unit. We prove that all these automorphisms are close to inner automorphisms.

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Abstract. Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry ( §1), the necessary material from algebraic logic ( §2), and a deeper introduction to the subject ( §3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3.In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry.Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras.§3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic.The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature.We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H.In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL).Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.

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.

The connection between classical model theoretical types (MT-types) and logically-geometrical types (LG-types) introduced by B. Plotkin is considered. It is proved that MT-types of two n-tuples in two universal algebras coincide if and only if their LG-types coincide. An algebra H is called logically perfect if for every two n-tuples in H whose types coincide, one can be sent to another by means of an automorphism of this algebra. Some sufficient condition for logically perfectness of free finitely generated algebras is given which helps to prove that finitely generated free Abelian groups, finitely generated free nilpotent groups and finitely generated free semigroups are logically perfect. It is proved that if two Abelian groups have the same type and one of them is finitely generated and free then these groups are isomorphic.