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

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.