Algebraic geometry over algebraic structures. IX: Principal universal classes and dis-limits. (English. Russian original) Zbl 1429.08007
Algebra Logic 57, No. 6, 414-428 (2019); translation from Algebra Logika 57, No. 6, 639-661 (2018).
Summary: 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.
For the Parts VIII and X, see [the authors, Fundam. Prikl. Mat. 22, No. 4, 75–100 (2019; Zbl 1470.08001); Int. J. Algebra Comput. 28, No. 8, 1425–1448 (2018; Zbl 1404.08006)].
For the Parts VIII and X, see [the authors, Fundam. Prikl. Mat. 22, No. 4, 75–100 (2019; Zbl 1470.08001); Int. J. Algebra Comput. 28, No. 8, 1425–1448 (2018; Zbl 1404.08006)].
MSC:
| 08A99 | Algebraic structures |
| 14A22 | Noncommutative algebraic geometry |
| 03C05 | Equational classes, universal algebra in model theory |
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 equivalenceReferences:
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