Quasiresolvent Models

Khisamiev, A. N.

doi:10.1023/b:allo.0000044284.10332.f9

Cited by 4 publications

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“…In [7], it was in fact proved that if M is an infinite model of a countably categorical theory, then the uniformization principle is violated in HF(M), and so such models do not exist for transitions 0, 1, and 9. For transitions 2, 4, 6, and 10, we can choose models of decidable countably categorical theories.…”

Section: Definition 16 (Quasiprojectibility)mentioning

confidence: 99%

Descriptive properties on admissible sets

Puzarenko

2010

Algebra Logic

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Relations are treated between the following descriptive properties on admissible sets: enumerability, uniformization, reduction, separation, extension. Moreover, in the setting of these properties, we consider existence problems for a universal computable function and for a computable function universal for {0; 1}-valued computable functions. It is shown that all relations between the given properties are strict. Also we look into algorithmic complexity of admissible sets lending support to the specified relations. It is stated that the reduction principle fails in some admissible sets over classical structures.For the majority of approaches to computability, a class of computably enumerable sets, which, as a rule, form a lattice with respect to set-theoretic operations, is not closed under complements. (Moreover, elements that have complements are exactly computable sets.) This necessitates the study of additional set-theoretic properties of the given class. In the present paper, we look at such properties for admissible sets. It will be shown that admissible sets behave rather badly in relation to much used properties. Namely, every relation between the properties may be realized on a certain admissible set, and in most cases, such a set can be chosen as a hereditarily finite superstructure over a computable model.

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Section: Definition 16 (Quasiprojectibility)mentioning

confidence: 99%

Descriptive properties on admissible sets

Puzarenko

2010

Algebra Logic

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“…Some version of the following proposition in the case of the logic action has been already appeared in [13]. …”

Section: Now (2) Is Easymentioning

confidence: 99%

Polish group actions, nice topologies, and admissible sets

Majcher-Iwanow¹

2008

Mathematical Logic Qtrly

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This is a continuation of [1]. We introduce the concept of a primarily quasiresolvent periodic abelian group and describe primarily quasiresolvent and 1-quasiresolvent periodic abelian groups.

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“…Quasiresolvent and 1-quasiresolvent models were defined in [1, 3] by analogy; see Definitions A and B below. Informally speaking, for a model to be quasiresolvent means that it has an effective representation as an increasing sequence of submodels which is called a quasiresolvent of the model in question.In this regard we are interested in finding some rich classes of quasiresolvent models among classical algebras that admit some "good" description.The quasiresolvent abelian p-groups and Ershov algebras were described in [1]. The goal of this article is to find broad classes of quasiresolvent periodic abelian groups.…”

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On quasiresolvent periodic abelian groups

Khisamiev

2007

Sib Math J

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This is a continuation of [1]. We introduce the concept of a primarily quasiresolvent periodic abelian group and describe primarily quasiresolvent and 1-quasiresolvent periodic abelian groups. We construct an example of a quasiresolvent but not primarily quasiresolvent periodic abelian group. For a direct sum of primary cyclic groups we obtain criteria for a group to be quasiresolvent, 1-quasiresolvent, and resolvent, and establish relations among them. We construct a set S of primes such that the direct sum of some cyclic groups of orders p ∈ S is not a quasiresolvent group.Ershov introduced the important concept of a quasiresolvent admissible set and proved in [2] that every hereditarily finite admissible set over a model of a regular theory is quasiresolvent. He also established there that every quasiresolvent admissible set admits a universal Σ-function and satisfies the reduction principle. Quasiresolvent and 1-quasiresolvent models were defined in [1, 3] by analogy; see Definitions A and B below. Informally speaking, for a model to be quasiresolvent means that it has an effective representation as an increasing sequence of submodels which is called a quasiresolvent of the model in question.In this regard we are interested in finding some rich classes of quasiresolvent models among classical algebras that admit some "good" description.The quasiresolvent abelian p-groups and Ershov algebras were described in [1]. The goal of this article is to find broad classes of quasiresolvent periodic abelian groups. In order to solve this problem we introduce some new classes of quasiresolvent groups by imposing various natural restrictions on quasiresolvents.In this article we give sufficient conditions for a model to be quasiresolvent (Proposition 1) as well as a necessary condition for a periodic abelian group to be quasiresolvent (Theorem 1). We introduce the concepts of a primarily quasiresolvent and G-quasiresolvent periodic abelian group (Definitions 1 and 2). For a group to be G-quasiresolvent means that there exists some quasiresolvent whose every step is a subgroup, while for a group to be primary quasiresolvent means that for every prime number p each step either meets the p-primary component trivially or includes it. Using these concepts we were able to obtain very simple descriptions of the classes of primarily quasiresolvent periodic abelian groups and G-quasiresolvent countable reduced periodic abelian groups in terms of sets of numbers. Therefore, we found a rather rich class of quasiresolvent periodic abelian groups. As corollaries to those results, we obtain (1) the model completeness of a quasiresolvent periodic abelian group (Corollary 2); (2) a criterion for a periodic abelian group to be 1-quasiresolvent (Corollary 3); (3) an example of a G-quasiresolvent but not primarily quasiresolvent periodic abelian group (Corollary 6); (4) a criterion for the direct sum of cyclic groups of distinct prime orders to be quasiresolvent, 1-quasiresolvent, or resolvent (Corollaries 7-10); and (5) the fact that the n...

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Quasiresolvent Models

Cited by 4 publications

References 0 publications

Descriptive properties on admissible sets

Descriptive properties on admissible sets

Polish group actions, nice topologies, and admissible sets

On quasiresolvent periodic abelian groups

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