Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set $\pi(G)$ of prime divisors of the profinite period of a group $G$ is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set $\pi(G)$ gives rise to equational Noetherianness. The concept of a standard linear pro-$p$-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-$p$-groups and free metabelian pro-$p$-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-$p$-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.