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

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

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

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

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

Let $a$ be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on $a$, so that for every $\Sigma^{-1}_a$--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 $\omega$; 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 $\Sigma^{-1}_a$--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 $\omega + \omega$; 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 $n\in\omega$, or $n=\omega$, and every notation of a nonzero ordinal there exists a $\Sigma^{-1}_a$--computable family of cardinality $n$, whose Rogers semilattice consists of exactly one element