id: 03926897 dt: j an: 03926897 au: Welch, Lawrence V. ti: A hierarchy of families of recursively enumerable degrees. so: J. Symb. Log. 49, 1160-1170 (1984). py: 1984 pu: Association for Symbolic Logic, Poughkeepsie, NY la: EN cc: ut: ci: li: doi:10.2307/2274268 ab: Assume that e is a number, b is a r.e. degree, and A is a set. We say that e is a representative for b in A if $e\in A$ and $W\sb e\in b$. Let A be a set and let ${\frak A}$ be a family of r.e. degrees. We say that A is a basis for ${\frak A}$ if the following two conditions hold: (1) every e in A is a representative in A for some b in ${\frak A}$; (ii) every b in ${\frak A}$ has a representative in A. Given any number n, we say that ${\tilde Δ}\sb n$ is the class of all ${\frak A}$ which have a $Δ\sb n$ basis, ${\tilde Π}\sb n$ is the class of all ${\frak A}$ which have a $Π\sb n$ basis, and ${\tilde Σ}\sb n$ is the class of all ${\frak A}$ which have a $Σ\sb n$ basis. In this paper it is shown that for all n, ${\tilde Δ}\sb n=Σ\sb n$ and for any n, ${\tilde Σ}\sb n\cup {\tilde Π}\sb n={\tilde Π}\sb n$ and ${\tilde Σ}\sb n\cap {\tilde Π}\sb n={\tilde Δ}\sb n$. The main results of this paper are the following theorems: Theorem 6. For all n, if ${\frak A}$ is ${\tilde Δ}\sb{n+1}$ then ${\frak A}$ is ${\tilde Π}\sb n$. Theorem 10. If $n\ge 3$ then ${\tilde Π}\sb n-{\tilde Π}\sb{n-1}\ne \emptyset$ (${\tilde Π}\sb 0={\tilde Π}\sb 1={\tilde Π}\sb 2)$. Theorem 11. If a is a Turing degree such that $a\ge 0\sp{(3)}$ then there is a family ${\frak A}$ such that deg(ind(${\frak A}))=a$. Theorem 12. If ${\frak A}$ is a nonempty family of r.e. degrees such that 0’ is not in ${\frak A}$, then $Rec\le\sb 1ind({\frak A})$. Theorem 15. If ${\frak A}$ is a nonempty family of r.e. degrees such that 0 is not in ${\frak A}$, and if ${\frak A}$ is ${\tilde Π}\sb 0$, then $Rec\le\sb 1ind({\frak A})$. rv: R.Sh.Omanadze (Tbilisi)