Abstract

We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an m-in- complete computably enumerable set A such that every set in S is m-reducible to A. For example, we show that the class of eectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded iff B is low 2 . For r = bwtt, tt, wtt and T , there is a bounded class intersecting every computably enumerable r-degree; for r = c, d and p, no such class exists. AMS Classication: 03D30; 03D25 Keywords: Computably enumerable sets (= Recursively enumerable sets); Simple sets; m-reducibility; Strong reducibilities; 3 classes; Ideals; Exact pairs 1 Introduction With a typical priority argument, one can show that for any simple set A, there is a simple set B such that B m A. Carl ...

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