The concept of n-cylinder and its relationship to simple sets

This function / is called the ^-cylinder function for P. It can be seen that a semicylinder is a 1-cylinder. In [5], properties of ^-cylinders and their relationship to cylinders were explored, and subsequently it was shown that: (i) for each n > 1, the class of all (n+ l)-cylinders is a proper subset of the class of all cylinders, and (ii) a set is a cylinder if and only if it is an ^-cylinder for each n>\. Thus we can deduce that as n tends to infinity, the class of-all ^-cylinders coincides with the class of all cylinders. This article shows a major difference between the properties of /^-cylinders and cylinders: for each n > 1, the class of all ^-cylinders contains a simple set whereas it has been shown [3] that no cylinder can be simple. The existence of ^-cylinders which are simple gives rise to the following question: "Can every one-one degree be represented by an ^-cylinder for each n > 1?" From the results obtained in [4] and [5] we conjecture that for every infinite recursively enumerable set We' there is an ^-cylinder Ane for each n > 1 such that We < An e and An e < We, where a set A is n* reducible to a 1-1 ' ' n*