Rogers Semilattices for Families of Equivalence Relations in the Ershov Hierarchy

[摘要] The paper studies Rogers semilattices for families of equivalence relations in the Ershov hierarchy. For an arbitrary notation a of a nonzero computable ordinal, we consider \(\sum\nolimits_a^{- 1} {}\)-computable numberings of the family of all \(\sum\nolimits_a^{- 1} {}\) equivalence relations. We show that this family has infinitely many pairwise incomparable Friedberg numberings and infinitely many pairwise incomparable positive undecidable numberings. We prove that the family of all c.e. equivalence relations has infinitely many pairwise incomparable minimal nonpositive numberings. Moreover, we show that there are infinitely many principal ideals without minimal numberings.

[学科分类] 数学（综合）

[关键词] Rogers semilattice;Ershov hierarchy;equivalence relation;computable numbering;Friedberg numbering;minimal numbering;universal numbering;principal ideal