Universal Algebraic Geometry

Universal algebraic geometry (another name is algebraic geometry over algebraic systems ^[1] ) is a direction in mathematics that studies the connections between elements of an algebraic system , expressed in the language of algebraic equations over algebraic systems . Classical algebraic geometry is a specific example of algebraic geometry over algebraic systems for the case of an algebraic field ; in the universal case, the tools of universal algebra are used to generalize the classical results.

The initial development of the direction was obtained in the works of Plotkin , Baumslag ( Eng. Gilbert Baumslag ), Kharlampovich , Myasnikov , Remeslennikov ^[2] . The starting point was the development of algebraic geometry over a free non-Abelian group; subsequently, substantial theories were obtained for rigid soluble groups ( Romanovsky ), metabelian groups , partially commutative groups , a number of results were revealed over abelian groups , topological groups , hyperbolic groups , ring algebras , and over a number of structures with a high level of generality, such as a semigroup , monoid , semilattice .

One of the main tasks of the direction is to describe algebraic sets over the chosen algebraic system ^[3] . The fundamental part of the theory is a generalization of the results of building algebraic geometry over specific types of algebraic systems and applying model-theoretic tools to construct similar theories over algebraic systems of any signature , finding common constructions that are not dependent on specific types of varieties of algebraic systems , selecting properties that are expressed regardless of types of manifolds and the identification of results that are universal for any systems of corresponding properties. One example of such a property is the Noetherness, previously developed separately for groups , rings , modules , but generalized for arbitrary algebraic systems, while for the entire class of non-first algebraic systems, there is a number of algebraic-geometric results. In addition to the universalization of the results, one of the technical effects of the approach is the simplification of many proofs due to the transition to a model-theoretic language that does not require the use of specific properties of groups, rings, modules.