@inbook{92ae1877001747b9ae16eb03a5735aab,

title = "Rogers semilattices in the analytical hierarchy: The case of finite families",

abstract = " 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 $S\subset P(\omega)$. 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. ",

keywords = "math.LO, 03D45, 03D55",

author = "Nikolay Bazhenov and Manat Mustafa",

note = "26 pages",

year = "2020",

month = oct,

day = "2",

language = "English",

series = "ALC Proccedings",

booktitle = "ALC2019 Procceding",

}