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This article is cited in 4 scientific papers (total in 4 papers)
Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits
E. Yu. Daniyarovaa, A. G. Myasnikovb, V. N. Remeslennikova a Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Schaefer School of Engineering and Science, Dep. of Math. Sci., Stevens Institute of Technology, Castle Point on Hudson, Hoboken NJ 07030-5991, USA
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 $\mathcal{A}$, 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.
Received: 06.02.2017 Revised: 10.10.2017
Citation:
E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits”, Algebra Logika, 57:6 (2018), 639–661; Algebra and Logic, 57:6 (2019), 414–428
Linking options:
https://www.mathnet.ru/eng/al872 https://www.mathnet.ru/eng/al/v57/i6/p639
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