Families Without Minimal Numberings

Abeshev, Kuanysh; Badaev, S. A.; Mustafa, Manat

doi:10.1007/s10469-014-9290-9

Cited by 4 publications

(1 citation statement)

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“…In view of the properties of the F. Stephan operator [17], it suffices to research Rogers semilattices for families of sets at two lower levels in the Ershov hierarchy. Other results on Rogers semilattices in Ershov hierarchy can be found, for example, in [18][19][20][21][22][23][24][25][26].…”

Section: математические наукиmentioning

confidence: 99%

On the Existence of Universal Numberings

Nurlanbek

2023

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The paper is devoted to research existence property of universal numberings for different computable families. A numbering α is reducible to a numbering β if there is computable function ƒ such that α = β ◦ ƒ. A computable numbering α for some family S is universal if any computable numbering β for the family S is reducible to α. It is well known that the family of all computably enumerable (c.e.) sets has a computable universal numbering. In this paper, we study families of almost all c.e. sets, recursive sets, and almost all differences of c.e. sets, namely questions about the existence of universal numberings for given families. We proved that there is no universal numbering for the family of all recursive sets. For families of c.e. sets without an empty set or a finite number of finite sets, there still exists a universal numbering. However, for families of all c.e. sets without an infinite set, there is no universal numbering. Also, we proved that family ∑2-1 \ Β and the family ∑1-1 has no universal ∑2-1-computable numbering for any Β ∈ ∑2-1.

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