### Abstract

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 A, i.e., algebraic structures in which all irreducible coordinate algebras over 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.

Original language | English (US) |
---|---|

Pages (from-to) | 414-428 |

Number of pages | 15 |

Journal | Algebra and Logic |

Volume | 57 |

Issue number | 6 |

DOIs | |

State | Published - Jan 15 2019 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Logic

### Cite this

*Algebra and Logic*,

*57*(6), 414-428. https://doi.org/10.1007/s10469-019-09514-6

}

*Algebra and Logic*, vol. 57, no. 6, pp. 414-428. https://doi.org/10.1007/s10469-019-09514-6

**Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits.** / Daniyarova, E. Y.U.; Miasnikov, Alexei; Remeslennikov, V. N.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits

AU - Daniyarova, E. Y.U.

AU - Miasnikov, Alexei

AU - Remeslennikov, V. N.

PY - 2019/1/15

Y1 - 2019/1/15

N2 - 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 A, i.e., algebraic structures in which all irreducible coordinate algebras over 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.

AB - 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 A, i.e., algebraic structures in which all irreducible coordinate algebras over 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.

UR - http://www.scopus.com/inward/record.url?scp=85063965160&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063965160&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s10469-019-09514-6

DO - https://doi.org/10.1007/s10469-019-09514-6

M3 - Article

VL - 57

SP - 414

EP - 428

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 6

ER -