Σ-Uniform structures and Σ-functions. I

Khisamiev, A. N.

doi:10.1007/s10469-011-9155-4

Algebra Logic

2011

DOI: 10.1007/s10469-011-9155-4

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Σ-Uniform structures and Σ-functions. I

A. N. Khisamiev

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Cited by 7 publications

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“…In [5,6], it was proved that a universal Σ-function exists in hereditarily finite superstructures over an Abelian p-group, a linear order, and an Ershov algebra. In [7,8], the concept of a Σ-uniform structure was introduced and a condition was specified that is necessary and sufficient for a universal Σ-function to exist * …”

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Universal Functions Over Trees

Khisamiev

2015

Algebra Logic

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Presented by the Program Committee of the Conference "Mal'tsev Readings"An important generalization of the notion of computability is Σ-definability (generalized computability) in admissible sets. This generalization made it possible to study computability problems over arbitrary structures, for example, over the field of reals. The most significant results in computability theory for admissible sets, as well as their applications in theoretical computer science (semantic programming, dynamic logic, the theory of effective f -spaces, and so on), are collected in [1]. There, also, the importance of the following direction for future research was pointed out: For a better understanding of the general nature of computability (constructive cognoscibility), it is necessary to further develop (gain a better insight into) the notion of computability in admissible sets of the form HF(A), a hereditarily finite superstructure over a structure A, where A either is a model of a rather simple theory or is one of the classical objects, such as, for instance, the field R of real numbers [1, p. 12].A fundamental result in classical computability theory is the existence of a universal partial computable function. We know from [1] that every admissible set of a finite signature contains a universal Σ-predicate, which is untrue for Σ-functions. A structure M, such that in the hereditarily finite superstructure HF(M) there exists no universal Σ-function, was constructed in [2]. In this connection, it is of interest to find a condition which guarantees, for a structure M, the existence of a universal Σ-function in the hereditarily finite superstructure HF(M).A review of further results on this problem can be found in [3]. In [4], a torsion-free Abelian group A was constructed for which HF(A) has no universal Σ-function. In [5,6], it was proved that a universal Σ-function exists in hereditarily finite superstructures over an Abelian p-group, a linear order, and an Ershov algebra. In [7,8], the concept of a Σ-uniform structure was introduced and a condition was specified that is necessary and sufficient for a universal Σ-function to exist *

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Universal Functions Over Trees

Khisamiev

2015

Algebra Logic

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“…The present paper is a continuation of [1], in which we introduced the concept of a Σ-uniform structure and derived a condition that is necessary and sufficient for a universal Σ-function to exist in a hereditarily finite admissible set over a Σ-uniform structure. Here we show how these results apply to Abelian groups and rings.…”

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“…We will adhere to the notation and terminology created for admissible sets in [2], for groups in [3], and for rings in [4] (see also [1]). …”

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