An example of a finite number of recursively enumerable $m$-degrees containing an infinite sequence of recursively enumerable sets

Abstract

The aim of this paper is to construct for an arbitrary natural number $n$ a recursively enumerable (r.e.) set $A$ such that the sets $A,A^2...,A^n$ belong to different $m$-degrees and the sets $A^n, A^{n+1}, A^{n+2},..$ belong to the same $m$-degree. That will provide an example of $n$ distinct r.e. $m$-degrees $d_m(A), d_m(A^2),..,d_m(A^n)$ belonging to one r.e. $btt$-degree (more precisely, $c$-degree, the degree $d_c(A)$ of the set $A$ coincidental with $d_c(A^2),..,d_c(A^n),..$). It suffices to prove that the set $A^{n+1}$ is $m$-reducible to the set $A^{n-1}$. For this purpose we construct a scheme for building a r.e. set $A$, corresponding to a sequence of r.e. sets $A_0,A_1,A_2,..,$ such that the first condition holds regardless the choise of $A_0,A_1,A_2,..,$. Further we build the sets $A_0,A_1,A_2,..,$ by steps, using the priority arguments to ensure the second condition.