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Algebra and Logic

, Volume 57, Issue 6, pp 414–428 | Cite as

Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits

  • E. Y. U. DaniyarovaEmail author
  • A. G. Myasnikov
  • V. N. Remeslennikov
Article
  • 8 Downloads

This paper enters into a series of works on universal algebraic geometry—a branch of mathematics that is presently flourishing and is still undergoing active development. The theme and subject area of universal algebraic geometry have their origins in classical algebraic geometry over a field, while the language and almost the entire methodological apparatus belong to model theory and universal algebra. The focus of the paper is the problem of finding Dis-limits for a given algebraic structure \( \mathcal{A} \), i.e., algebraic structures in which all irreducible coordinate algebras over \( \mathcal{A} \) are embedded and in which there are no other finitely generated substructures. Finding a solution to this problem necessitated a good description of principal universal classes and quasivarieties. The paper is divided into two parts. In the first part, we give criteria for a given universal class (or quasivariety) to be principal. In the second part, we formulate explicitly the problem of finding Dis-limits for algebraic structures and show how the results of the first part make it possible to solve this problem in many cases.

Keywords

universal algebraic geometry algebraic structure universal class quasivariety joint embedding property irreducible coordinate algebra discriminability Dis-limit equational Noetherian property equational codomain universal geometric equivalence 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • E. Y. U. Daniyarova
    • 1
    Email author
  • A. G. Myasnikov
    • 2
  • V. N. Remeslennikov
    • 1
  1. 1.Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of SciencesOmskRussia
  2. 2.Schaefer School of Engineering and Science, Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on HudsonHobokenUSA

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