For every variety of algebras and every algebras in these variety we can
consider an algebraic geometry. Algebras may be many sorted (not necessarily
one sorted) algebras. A set of sorts is fixed for each variety. This theory can
be applied to the variety of representations of groups over fixed commutative
ring with unit. We consider a representation as two sorted algebra. We
concentrate on the case of the action type algebraic geometry of
representations of 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.Comment: 35 page
