TY - JOUR
T1 - Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits
AU - Daniyarova, E. Y.U.
AU - Myasnikov, A. G.
AU - Remeslennikov, V. N.
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
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.
KW - Dis-limit
KW - algebraic structure
KW - discriminability
KW - equational Noetherian property
KW - equational codomain
KW - irreducible coordinate algebra
KW - joint embedding property
KW - quasivariety
KW - universal algebraic geometry
KW - universal class
KW - universal geometric equivalence
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U2 - 10.1007/s10469-019-09514-6
DO - 10.1007/s10469-019-09514-6
M3 - Article
AN - SCOPUS:85063965160
SN - 0002-5232
VL - 57
SP - 414
EP - 428
JO - Algebra and Logic
JF - Algebra and Logic
IS - 6
ER -