G-IDENTITIES AND G-VARIETIES

The fundamentals of algebraic geometry over a fixed group G were propounded in [1] where, in particular, the notion of a category of G-groups is introduced. Again, for groups in this category; the concepts of a G-identity and of a G-variety can be defined. We outline the groundwork for the theory of varieties in the category of G-groups. Most essential here is the idea of a group Vn,red(G) - of reduced G-identities of rank n. which has a strong impact on the computations of coordinate, groups for algebraic sets over G. We prove that Vn,red(G) = I for all natural n if G is a free-like or relatively free group for some variety of nilpotent groups whose rank is not less than the nilpotency class of G.