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

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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## Preview

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## References

- 1.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over algebraic structures. II. Foundations,”
*Fundam. Prikl. Mat.*,**17**, No. 1, 65-106 (2011/2012).Google Scholar - 2.V. A. Gorbunov,
*Algebraic Theory of Quasivarieties, Sib. School Alg. Log.*[in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar - 3.S. V. Sudoplatov,
*Classification of Countable Models of Complete Theories*, Part 1, Novosibirsk, Novosibirsk State Tech. Univ. (2014).Google Scholar - 4.S. V. Sudoplatov,
*Classification of Countable Models of Complete Theories*, Part 2, Novosibirsk, Novosibirsk State Tech. Univ. (2014).Google Scholar - 5.G. Baumslag, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over groups. I: Algebraic sets and ideal theory,”
*J. Alg.*,**219**, No. 1, 16-79 (1999).MathSciNetCrossRefzbMATHGoogle Scholar - 6.O. Kharlampovich and A. Myasnikov, “Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and Nullstellensatz,”
*J. Alg.*,**200**, No. 2, 472-516 (1998).MathSciNetCrossRefzbMATHGoogle Scholar - 7.O. Kharlampovich and A. Myasnikov, “Irreducible affine varieties over a free group. II: Systems in triangular quasi-quadratic form and description of residually free groups,”
*J. Alg.*,**200**, No. 2, 517-570 (1998).MathSciNetCrossRefzbMATHGoogle Scholar - 8.W. Hodges,
*Model Theory, Enc. Math. Appl.*, 42, Cambridge Univ. Press, Cambridge (1993).Google Scholar - 9.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Unification theorems in algebraic geometry,” in
*Aspects of Infinite Groups, Algebra Discr. Math. (Hackensack)*, 1, World Sci., Hackensack, NJ (2008), pp. 80-111.Google Scholar - 10.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over algebraic structures. III: Equationally Noetherian property and compactness,”
*South. Asian Bull. Math.*.**35**, No. 1, 35-68 (2011).MathSciNetzbMATHGoogle Scholar - 11.E. Yu. Daniyarova, A. G. Myasnikov, and V. N. Remeslennikov, “Algebraic geometry over algebraic structures. IV. Equational domains and codomains,”
*Algebra and Logic*,**49**, No. 6, 483-508 (2010).MathSciNetCrossRefzbMATHGoogle Scholar - 12.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over algebraic structures. V. The case of arbitrary signature,”
*Algebra and Logic*,**51**, No. 1, 28-40 (2012).MathSciNetCrossRefzbMATHGoogle Scholar - 13.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over algebraic structures. VI. Geometrical equivalence,”
*Algebra and Logic*,**56**, No. 4, 281-294 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 14.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Universal geometrical equivalence of the algebraic structures of common signature,”
*Sib. Math. J.*,**58**, No. 5, 801-812 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 15.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures,” to appear in
*Fund. Prikl. Mat.*Google Scholar - 16.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Universal algebraic geometry,”
*Dokl. Akad. Nauk*,**439**, No. 6, 730-732 (2011).MathSciNetzbMATHGoogle Scholar - 17.E. Daniyarova, A. Myasnikov, and V. Remeslennikov, “Dimension in universal algebraic geometry,”
*Dokl. Ross. Akad. Nauk*,**457**, No. 3, 265-267 (2014).MathSciNetzbMATHGoogle Scholar - 18.A. I. Mal’tsev,
*Algebraic Systems*[in Russian], Nauka, Moscow (1970).Google Scholar - 19.A. A. Mishchenko, V. N. Remeslennikov, and A. V. Treier, “Universal invariants for classes of Abelian groups,”
*Algebra and Logic*,**56**, No. 2, 116-132 (2017).MathSciNetCrossRefzbMATHGoogle Scholar