An example on existence of a 02n-computable family of total functions whose rogers semilattice contains an ideal without minimal elements

Abstract

We study computable families of total functions of any level of the Kleene-Mostowski hierarchy above level 1 and try to find elementary properties of the Rogers semilattices that are different from the properties of the classical Rogers semilattices for families of computable functions. It is known that on first level of the arithmetical hierarchy the Rogers semilattice of any computable family of total functions contains no ideal without minimal elements, [1]. In this article we show an example how to build 02n-computable family of total functions whose Rogers semilattice contains an ideal without minimal elements, n.