2 1 Ja n 20 05 ACTION TYPE GEOMETRICAL EQUIVALENCE OF REPRESENTATIONS OF GROUPS . May 4 , 2008

For every variety of algebras Θ 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 V. We concentrate on the case of the action type algebraic geometry of representations of 1 groups. In this case algebraic sets are defined by systems of action type equations and equations in the acting group are not considered. This is the special case, which cannot be deduced from the general theory. In this paper the following basic notions are introduced: action type geometrical equivalence of two representations, action type quasi-identity in representations, action type quasi-variety of representations , action type Noetherian variety of representations, action type geometrically Noetherian representation, action type logically Noetherian representation. Proposition 6.2, and Corollary from Proposition 6.3 provide examples of action type Noetherian variety of representations and action type geometrically Noetherian representations. In Corollary 2 from Theorem 5.1 the approximation-like criterion for two representations to be action type geometrically equivalent is proved. This criterion is similar to the approximation criterion for two algebras to be geometrically equivalent in regular sense ([PPT]). Theorem 6.2 gives a criterion for a representation to be action type logically Noetherian. This criterion is formulated in terms of an action type quasi-variety generated by a representation (compare with [Pl4]). In Corollary 2 from Theorem 7.1 we consider a Birkhoff-like description [Bi] of an action type quasi-variety generated by a class of representations. Theorem 8.1 shows that: there exists a continuum of non isomorphic simple modules over KF 2 , where F 2 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, in Section 9 we give an example of a non action type logically Noetherian representation. This allows to build an ultrapower of a non action type logically Noetherian representation, which has the same action type quasi-identities but is not action type geometrically equivalent to the original representation ((Corollary from Theorem 9.1). This result is parallel to the corresponding theorem for …