TY - JOUR

T1 - Rogers semilattices of families of two embedded sets in the Ershov hierarchy

AU - Badaev, Serikzhan A.

AU - Manat, Mustafa

AU - Sorbi, Andrea

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/8

Y1 - 2012/8

N2 - Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every -computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On the other hand, we also give a sufficient condition on a, that yields that there is a -computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a -computable family of cardinality n, whose Rogers semilattice consists of exactly one element.

AB - Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every -computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On the other hand, we also give a sufficient condition on a, that yields that there is a -computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a -computable family of cardinality n, whose Rogers semilattice consists of exactly one element.

KW - Computable numbering

KW - Computable ordinal

KW - Ershov's hierarchy

KW - Ordinal notation

KW - Rogers semilattice

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U2 - 10.1002/malq.201100114

DO - 10.1002/malq.201100114

M3 - Article

AN - SCOPUS:84864767814

VL - 58

SP - 366

EP - 376

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

SN - 0942-5616

IS - 4-5

ER -