SOME RESULTS AND PROBLEMS RELATED TO UNIVERSAL ALGEBRAIC GEOMETRY

Abstract

In universal algebraic geometry (UAG), some primary notions of classical algebraic geometry are applied to an arbitrary variety of algebras Θ and an arbitrary algebra H ∈ Θ. We consider an algebraic geometry in Θ over the distinguished algebra H and we also analyze H from the point of view of its geometric properties. This insight leads to a system of new notions and stimulates a number of new problems. They are new with respect to algebra, algebraic geometry and even with respect to the classical algebraic geometry. In our approach, there are two main aspects: the first one is a study of the algebra H and its geometric properties, while the second is focused on studying algebraic sets and algebraic varieties over a "good", particular algebra H. Considering the subject from the second standpoint, the main goal is to get forward as far as possible in a classification of algebraic sets over the given H. The first approach does not require such a classification which is itself an independent and extremely difficult task. We also consider some geometric relations between different H 1 and H 2 in Θ. The present paper should be viewed as a brief review of what has been done in universal algebraic geometry. We also give a list of unsolved problems for future work.