Supersimple sets and the problem of extending a retracing function

Abstract

An infinite set A of natural numbers is called regressive if there is a (non-repeating) sequential ordering 0 , CLU " of A and a partial recursive function / such that A <i domain (/), /(o) = #0, and (V%)[/( n +i) = &J; in case o > du * * * is the natural ordering of A by increasing size of elements, A is called retraceable and / is said to retrace A. It sometimes happens that a retraceable set does not admit an everywhere-defined retracing function, or that it does not admit a finite-to-one retracing function. Question: does there exist an infinite set A of natural numbers such that A is retraced both by a total recursive function and by a finite-to-one partial recursive function, but not by a function which is both total recursive and finite-toone? Via some theorems relating to D. A. Martin's notions of supersimple and superimmune sets of natural numbers, a strongly affirmative result is obtained.