Classes bounded by incomplete sets

Abstract We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We say that a class S of computably enumerable sets bounded iff there is an m -incomplete computably enumerable set A such that every set in S is m -reducible to A . For example, we show that the class of effectively 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.