Approximations of theories

Sudoplatov, S. V.

doi:10.33048/semi.2020.17.049

Cited by 11 publications

(27 citation statements)

References 17 publications

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“…It was shown in [6] that any e-minimal family T has unique accumulation point T with respect to neighbourhoods T ϕ , and T ∪ {T } is also called e-minimal.…”

Section: Preliminariesmentioning

confidence: 99%

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Approximations for Theories of Abelian Groups

Pavlyuk¹,

Sudoplatov²

2020

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Approximations of syntactic and semantic objects play an important role in various fields of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, infinite objects can be approximated by finite or strongly minimal ones. Thus, complicated objects can be collected using simplified ones. Among these objects, Abelian groups, their first order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of finite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.

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“…It was shown in [6] that any e-minimal family T has unique accumulation point T with respect to neighbourhoods T ϕ , and T ∪ {T } is also called e-minimal.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this paper we apply a general approach for approximations of theories [6] to natural families of theories of Abelian groups and describe some forms of approximability.…”

Section: Introductionmentioning

confidence: 99%

Approximations for Theories of Abelian Groups

Pavlyuk¹,

Sudoplatov²

2020

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“…Throughout the paper we consider families T of complete first-order theories of a relational language Σ = Σ(T ) and use the following terminology in [1,2,3,4,5,6,7,8,9,10].…”

Section: Preliminariesmentioning

confidence: 99%

“…It was shown in [8] that any e-minimal family T has unique accumulation point T with respect to neighbourhoods T φ , and T ∪ {T } is also called e-minimal.…”

Section: Preliminariesmentioning

confidence: 99%

Algebras for definable families of theories

Markhabatov¹,

Sudoplatov²

2019

Sib. Elektron. Mat. Izv.

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“…We continue to study families of theories [1,2,3,4,5,6] and their approximations [7] introducing ranks and degrees for families of theories, similar to Morley rank and degree [8], as well as Cantor-Bendixson rank and degree, and the notion of totally transcendental family of theories. These ranks and degree plays a similar role for families of theories, with hierarchies for definable sets of theories, as Morley ones for a fixed theory although they have own specificities.…”

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confidence: 99%

Ranks for families of theories and their spectra

Sudoplatov¹

2019

Preprint

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We define ranks and degrees for families of theories, similar to Morley rank and degree, as well as Cantor-Bendixson rank and degree, and the notion of totally transcendental family of theories. Bounds for e-spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under E-closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of E-closure and of the e-spectrum value, for a countable language, are proved.

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Approximations of theories

Cited by 11 publications

References 17 publications

Approximations for Theories of Abelian Groups

Approximations for Theories of Abelian Groups

Algebras for definable families of theories

Ranks for families of theories and their spectra

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