We are concerned with Ershov’s problem of obtaining a characterization of families with one-element Rogers semilattice (the semilattice of computable enumerations of a family). An algorithmic description is furnished for the families of recursive functions whose Rogers semilattice is one-element. It is proved that there exists a nontrivial family of recursively enumerable sets with the least set under inclusion, whose Rogers semilattice consists of a single element.
KeywordsRecursive Function Numeration Theory Trivial Consequence Computable Enumeration Construction Step
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- 1.S. S. Goncharov and S. A. Badaev, “Classes with pairwise equivalent enumerations,” inLogical Foundations of Computer Science, Lect. Notes Comp. Sc.,813, 140–141 (1994).Google Scholar
- 2.Yu. L. Ershov,Numeration Theory [in Russian], Nauka, Moscow (1977).Google Scholar
- 3.H. Rogers,Theory of Recursive Function and Effective Computability, McGraw-Hill, New York (1967).Google Scholar
- 4.J. L. Kelley,General Topology, Van Nostrand, New York (1957).Google Scholar
- 6.V. A. Uspenskii, “Systems of recursively enumerable sets and their enumerations,”Dokl. Akad. Nauk SSSR,105, No. 6 1155–1158 (1955).Google Scholar
- 7.Yu. L. Ershov, “Numerations of the families of general recursive functions,”Sib. Mat. Zh.,8, No. 5 1015–1025 (1967).Google Scholar
- 8.S. S. Goncharov, “Positive computable enumerations,”Dokl. Ross. Akad. Nauk,332, No. 2, 142–143 (1993).Google Scholar
- 9.S. S. Goncharov, “Equivalent constructivizations,” Doctoral Dissertation, Novosibirsk (1981).Google Scholar
- 13.A. I. Mal’tsev, “Positive and negative enumerations,”Dokl. Akad. Nauk SSSR,160, No. 2, 278–280 (1965).Google Scholar
- 14.V. V. Viugin, “Certain examples of upper semilattice of computable numerations,”Algebra Logika,12, No. 5, 512–529 (1973).Google Scholar
- 16.S. A. Badaev, “Cardinality of semilattices of enumerations of nondiscrete families,”Sib. Math. Zh.,34, No. 5, 3–10 (1993).Google Scholar