Autostable I-Algebras

shown that their computable dimension may assume only the value 1 or ω, and a complete characterization of computable categoricity was given.In the present paper, we study the question about spectrum of possible computable dimensions of trees enriched by an initial subtree (briefly, I-trees). It is proved that the computable dimension of any computable I-tree of infinite height is ω. Moreover, this dimension is effectively infinite, in the sense that, given any uniformly presented list of computable copies of the same I-tree, we can construct another computable copy of that tree, which is not computably isomorphic to any of the copies on the list. Notice that the results obtained can be naturally generalized to the case of several distinguished initial subtrees.

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“…, a r−1 . A proof of the previous theorem, offered in [6], implies that the present theorem admits the following modification.…”

Section: The Notation and Basic Definitionsmentioning

confidence: 80%

“…A number of generalizations and modifications of this method have been worked up to date (see [3]). We will need the following two versions of the theorem on branching models, the first of which was proven in [6].…”

Section: The Notation and Basic Definitionsmentioning

confidence: 99%

The Computable Dimension of I-Trees of Infinite Height

2004

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“…Finally, R is a Δ-ideal if R is simultaneously a Σ-and a Π-ideal. Note that the notion of Σ-ideals appeared earlier in the description of computably categorical I-algebras in [1], where such were called s-definable ideals.…”

Section: Relatively Intrinsically Computable Idealsmentioning

confidence: 99%

Computable ideals in I-algebras

Alaev

2010

Algebra Logic

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Keywords: Boolean algebra with finite number of distinguished ideals, intrinsically computable ideal, relatively intrinsically computable ideal.We give algebraic descriptions of relatively intrinsically computable ideals in I-algebras (Boolean algebras with a finite number of distinguished ideals) and of intrinsically computable ideals for the case of two distinguished ideals in the language of I-algebras.

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“…Examples of such methods are the branching theorem in [13] and the theorem on unbounded structures in [14]. Note that an essential generalization of the branching theorem mentioned was obtained in [15].…”

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confidence: 99%

“…The branching theorem is widely used in research on computably categorical structures of various natural classes such as linear orders, Boolean algebras [13], Abelian p-groups [14,16], atomic ideal enrichments of Boolean algebras [15,17,18], etc. The theorem on unbounded structures has been applied in studying Abelian groups of infinite rank [14], projective planes [19,20], Boolean algebras with distinguished automorphism, etc.…”

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confidence: 99%

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

Bazhenov

2015

Algebra Logic

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Computable categoricity in the Ershov hierarchy is studied. We consider F a -categorical and G a -categorical structures. These were introduced by B. Khoussainov, F. Stephan, and Y. Yang for a, which is a notation for a constructive ordinal. A generalization of the branching theorem is proved for F a -categorical structures. As a consequence we obtain a description of F a -categorical structures for classes of Boolean algebras and Abelian p-groups. Furthermore, it is shown that the branching theorem cannot be generalized to G a -categorical structures.Research on computable categoricity of structures was initiated by B. L. van der Waerden who studied the question whether there exists an effective algorithm for constructing an isomorphism between two different algebraic closures of a field built in different ways. Subsequently, starting with 1950s, computable categoricity of structures was explored by A. I. Mal'tsev, A. Fröhlich, J. Shepherdson, S. S. Goncharov, J. Remmel, C. Ash, J. Knight, and many other authors.In investigating computably categorical structures, one of the most important problems is to gain an insight into computable categoricity relative to classes of known hierarchies of sets. In particular, of special interest is the Ershov hierarchy introduced in [1-3]. There are quite many works dealing with the Ershov hierarchy. (Among these, it is worth mentioning resent papers [4-10] that focus primarily on numberings in the Ershov hierarchy.) Research on computable categoricity relative to levels of the Ershov hierarchy was started in [11]. There, for an arbitrary a, which is *

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Autostable I-Algebras

Cited by 11 publications

References 6 publications

The Computable Dimension of I-Trees of Infinite Height

The Computable Dimension of I-Trees of Infinite Height

Computable ideals in I-algebras

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

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