On Regressive Isols and Comparability of Summands and a Theorem of R. Downey

Abstract

In this paper we present a collection of results related to the comparability of summands property of regressive isols. We show that if an infinite regressive is01 has comparability of summands, then every predecessor of the isol has a weak comparability of summands property. Recently R. Downey proved that there exist regressive isols that are both hyper-torre and cosimple. There is a surprisingly close connection between nonrecursive recursively enumerable sets and particular retraceable sets and regressive isols. We apply the theorem of Downey to show that among the regressive isols that are related to recursively enumerable sets there are some with a new property.Mathematics Subject Classification: 03D50.

A Fine Structure in the Theory of Isols

“…This result was proved by R. DOWNEY in [7]. That this fact has some interesting connection with particular r. e. sets we tried to show in [3]. It seems like a nice direction to see if a cosimple completely torre is01 may exist.…”

“…Also, since T is a recursive tree and T n U and T n V are recursive sets, we see that there shall exist a uniformly effective method so that for any node p for which it is known that there are values as SO and s1 that make (5. 3) true and that are incomparable branching nodes which lie beyond p , a pair of first such values can be effectively computed from the value of p . Let us now assume that we have such a recursive procedure.…”

“…In addition, the sets S n U and S n V are recursive, and the relations like (5. 3) and (5.4) that are known to be true in T , now also.are true in S , since S is a subset of U U V . Hence, if i and j are any successive branching nodes of S and i 4 j , then (5.5) IU n (4.4 I 5 IV fl ( i , j l I.…”