Semilattices of Punctual Numberings

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The theory of numberings studies uniform computations for classes of mathematical objects. A large body of literature is devoted to investigations of computable numberings, i.e. uniform enumerations for families of computably enumerable sets, and the reducibility among these numberings. This reducibility, induced by Turing computable functions, aims to classify the algorithmic complexity of numberings. The paper is inspired by the recent advances in the area of punctual algebraic structures. We recast the classical studies of numberings in the punctual setting—we study punctual numberings, i.e. uniform computations for families of primitive recursive functions. The reducibility between punctual numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We prove that any infinite Rogers pr-semilattice is dense and does not have minimal elements. Furthermore, we give an example of infinite Rogers pr-semilattice, which is a lattice. These results exhibit interesting phenomena, which do not occur in the classical case of computable numberings and their semilattices.

Original languageEnglish
Title of host publicationTheory and Applications of Models of Computation - 16th International Conference, TAMC 2020, Proceedings
EditorsJianer Chen, Qilong Feng, Jinhui Xu
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages12
ISBN (Print)9783030592660
Publication statusPublished - 2020
Event16th Annual Conference on Theory and Applications of Models of Computation, TAMC 2020 - Changsha, China
Duration: 18 Oct 202020 Oct 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12337 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference16th Annual Conference on Theory and Applications of Models of Computation, TAMC 2020


  • Friedberg numbering
  • Numbering
  • Online computation
  • Primitive recursion
  • Punctual structure
  • Rogers semilattice
  • Upper semilattice


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