Rogers semilattices of families of two embedded sets in the Ershov hierarchy
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Date
2012
Authors
Badaev, Serikzhan A.
Mustafa, M.
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Abstract
Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a su cient condition on a, so that for every 1 a {computable family of two embedded sets, i.e. two sets A;B, with A properly contined in B, the Rogers semilattice of the family is in nite. This condition is satis ed by every notation of !; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satis es this condition. On the other hand, we also give a su cient condition on a,
that yields that there is a 1 a {computable family of two embedded sets,
whose Rogers semilattice consists of exactly one element; this condition is
satis ed by all notations of every successor ordinal bigger than 1, and by
all notations of the ordinal !+!; moreover every computable ordinal that
is sum of two smaller ordinals has a notation that satis es this condition.
We also show that for every nonzero n 2 !, or n = !, and every notation
of a nonzero ordinal there exists a 1 a {computable family of cardinality
n, whose Rogers semilattice consists of exactly one element.
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Keywords
Research Subject Categories::MATHEMATICS, Ershov hierarchy
Citation
Badaev Serikzhan A., Mustafa M.; 2012; Rogers semilattices of families of two embedded sets in the Ershov hierarchy