Algebraic geometry over algebraic structures. II. Foundations

Abstract

MSC 03C99; 08A99; 14A99In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove so-called Unification Theorems which describe coordinate algebras of algebraic sets in several different ways.

Equations in polyadic groups

“…We say that (G, f ) is equational noetherian, if for any system S of polyadic equations, there exists a finite subsystem S 0 such that V G (S) = V G (S 0 ). For general properties of equational noetherian algebraic systems, see [7]. One can define a topology on G m using algebraic sets as a prebase: in this topology every closed set is an arbitrary intersection of finite unions of algebraic sets.…”

“…It can be shown that Y 1 and Y 2 are isomorphic, if and only if, they have the same coordinate polyadic groups. Therefore, it is very important to determine the structure of polyadic groups which are the coordinate polyadic groups of algebraic sets in G. The unification theorems of [7] provide strong tools to do this. Here, we give two versions of the unification theorems which we can use to determine coordinate polyadic groups.…”

“…Here, we give two versions of the unification theorems which we can use to determine coordinate polyadic groups. For details, see [7].…”

“…This interesting subject is the result of decades of efforts of many algebraists to solve the classical problems of Alfred Tarski concerning the first order theory of nonabelian free groups (see [23], [24] and [25]). Equations over Lie algebras, associative algebras and semigroups are also studied (for example, see [7], [10], and [33]). In a series of fundamental papers, Daniyarova, Myasnikov and Remeslennikov introduced the basics of universal algebraic geometry ( [6], [7], [8] and [9]).…”

“…Our notations dealing with universal algebraic geometry, are the same as [6], [7], [8], [9]. The reader should consult these references for a complete account of the universal algebraic geometry.…”