Algebras for definable families of theories

Markhabatov, N. D.; Sudoplatov, S. V.

doi:10.33048/semi.2019.16.038

Cited by 5 publications

(3 citation statements)

References 6 publications

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“…Then either all E-classes have the same cardinality i.e. for any E-class there is an infinite number of isomorphic E-classes (1), or some finite number of E-classes have the same cardinality (2). Since all elements of each E-class are equivalent, there is an automorphism mapping this class onto itself.…”

Section: Resultsmentioning

confidence: 99%

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Approximations of the Theories of Structures With One Equivalence Relation

Markhabatov

2023

jour

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Recently, various methods similar to the “transfer principle” have been rapidly developing, where one property of a structure or pieces of this structure is satisfied in all infinite structures or in another algebraic structure. Such methods include smoothly approximable structures, holographic structures, almost sure theories, and pseudofinite structures approximable by finite structures. Pseudofinite structures are mathematical structures that resemble finite structures but are not actually finite. They are important in various areas of mathematics, including model theory and algebraic geometry. Pseudofinite structures are a fascinating area of mathematical logic that bridge the gap between finite and infinite structures. They allow studying infinite structures in ways that resemble finite structures, and they provide a connection to various other concepts in model theory. Further studying pseudofinite structures will continue to reveal new insights and applications in mathematics and beyond. Pseudofinite theory is a branch of mathematical logic that studies structures that are similar in some ways to finite structures, but can be infinitely large in other ways. It is an area of research that lies at the intersection of model theory and number theory and deals with infinite structures that share some properties with finite structures, such as having only finitely many elements up to isomorphism. A. Lachlan introduced the concept of smoothly approximable structures in order to change the direction of analysis from finite to infinite, that is, to classify large finite structures that seem to be smooth approximations to an infinite limit. The theory of pseudofinite structures is particularly relevant for studying equivalence relations. In this paper, we study the model-theoretic property of the theory of equivalence relations, in particular, the property of smooth approximability. Let L = {E}, where Е is an equivalence relation. We prove that an any ω-categorical L-structure M is smoothly approximable. We also prove that any infinite L-structure M is pseudofinite.

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Section: Resultsmentioning

confidence: 99%

“…Approximations for permutation theory by theories of finite structures were studied in [1], for locally free algebras in [2], for ordered theories in [3], for Abelian groups in terms of Szmielev invariants in [4,5], for regular graphs studied in [6].…”

Section: Introductionmentioning

confidence: 99%

Approximations of the Theories of Structures With One Equivalence Relation

Markhabatov

2023

jour

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“…Kudaibergenov introduced and studied the notions of a weakly quasi-ominimal model and theory [128] (2010). He introduced and studied various extensions of o-minimality into partial orders [130,135] [159,160,187,188,189]. In particular, B.Sh.…”

Section: B2 On the One Hand If For Somementioning

confidence: 99%