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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2025. Vol 54

Expansions and Restrictions of Structures and Theories, Their Hierarchies

Author(s)

Sergey V. Sudoplatov1, 2

1 Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russian Federation 

2 Novosibirsk State Technical University, Novosibirsk, Russian Federation

Abstract
The study and description of possibilities of expansions and restrictions of structures and their theories is used to obtain a structural information both in general and for various natural algebraic, geometric, ordered theories and models. The origins for the description are based on known model-theoretic operations of Morleyzation, or Atomization, and Skolemization, allowing to preserve or naturally extend formulaically definable sets of a given structure, and obtain a level of quantifier elimination, where formulaically definable sets are represented as Boolean combinations of definable sets specified by quantifier-free formulae. The operations of Shelahizations, or Namizations, produce both extensions and expansions of a structure giving names or labels for definable sets. In the paper, we introduce and study some general principles and hierarchical properties of expansions and restrictions of structures and their theories. These principles are based on upper and lower cones, lattices, and permutations. The general approach is applied to describe these properties for classes of 𝜔-categorical theories and structures, Ehrenfeucht theories and their models, strongly minimal, 𝜔1-categorical, and stable ones. Here all these classes are closed under permutations. It is proved that any fusions of strongly minimal structures are strongly minimal, too, whereas the properties of 𝜔-categoricity, Ehrenfeuchtness, 𝜔1-categoricity, and stability can fail under fusions. It is also shown that the classes of 𝜔-categorical, strongly minimal and stable regular structures are closed under lower cones of all their elements, whereas the classes of Ehrenfeucht and 𝜔1-categorical structures do not have that property, with some infinite chains of expansions alternating Ehrenfeuchtness and non-Ehrenfeuchtness, and other infinite chains alternating 𝜔1-categoricity and non-𝜔1-categoricity.
About the Authors
Sergey V. Sudoplatov, Dr. Sci. (Phys.-Math.), Prof., Sobolev Institute of Mathematics SB RAS, Novosibirsk, 630090, Russian Federation, sudoplat@math.nsc.ru; Novosibirsk State Technical University, Novosibirsk, 630073, Russian Federation, sudoplatov@corp.nstu.ru
For citation
Sudoplatov S. V. Expansions and Restrictions of Structures and Theories, Their Hierarchies. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 54, pp. 143–159. https://doi.org/10.26516/1997-7670.2025.54.143
Keywords
hierarchy, property, expansion of structure, restriction of structure, theory
UDC
510.67
MSC
03C30, 03C45, 03C52
DOI
https://doi.org/10.26516/1997-7670.2025.54.143
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