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تسجيل مستخدم جديدRogers semilattices in the analytical hierarchy: The case of finite families
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A numbering of a countable family $S$ is a surjective map from the set of natural numbers $omega$ onto $S$. The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings, for families $Ssubset P(omega)$. Working in set theory ZF+DC+PD, we obtain the following results on families from various levels of the analytical hierarchy. For a non-zero number $n$, by $E^1_n$ we denote $Pi^1_n$ if $n$ is odd, and $Sigma^1_n$ if $n$ is even. We show that for a finite family $S$ of $E^1_n$ sets, its Rogers $E^1_n$-semilattice has the greatest element if and only if $S$ contains the least element under set-theoretic inclusion. Furthermore, if $S$ does not have the $subseteq$-least element, then the corresponding Rogers $E^1_n$-semilattice is upwards dense.
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A numbering of a countable family $S$ is a surjective map from the set of natural numbers $omega$ onto $S$. A numbering $ u$ is reducible to a numbering $mu$ if there is an effective procedure which given a $ u$-index of an object from $S$, computes
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