On co-simple isols and their intersection types
type: Item
instance of: scholarly article
title: On co-simple isols and their intersection types
zbMATH Open document ID: 0773.03031
DOI: 10.1016/0168-0072(92)90074-A
author: Theodore A. Slaman
author: Rodney G. Downey
published in: Annals of Pure and Applied Logic
publication date: 1992-09-27T00:00:00Z
review text: In Pac. J. Math. 79, 19-35 (1978; Zbl 0404.03034), \textit{J. Barback} and \textit{T. G. McLaughlin} introduced the notion of intersection type in isol theory. Denoting the recursive equivalence type of \(A\) by \([A]\), the intersection type of two regressive isols \([A]\) and \([B]\) is defined to be the collection of all \([A'\cap B']\) such that \([A']=[A]\) and \([B']=[B]\). They showed: (*) for every co-simple regressive isol \([A]\) there exists a regressive isol \([B]\) such that the intersection type of the two is trivial, i.e., consists precisely of the finite isols. In Aspects of effective algebra, Proc. Conf., Monash Univ./Aust. 1979, 189-195 (1981; Zbl 0472.03036) and in his monograph ``Regressive sets and the theory of isols'' (1982; Zbl 0484.03025), \textit{T. G. McLaughlin} asked whether the \(\text{isol\;}[B]\) in (*) may be taken to be co-simple. The present paper answers that question in the affirmative, using ``a nonuniform \(0'''\) priority argument that can be viewed as the execution of a single strategy from a \(0^{(4)}\)-argument''. In his monograph, McLaughlin also asked whether \(B\) could be chosen to have the same Turing degree as \(A\). The authors answer this negatively by showing that in (*), \(A\times B\) cannot have low degree.
Mathematics Subject Classification ID: 03D50
zbMATH DE Number: 60752
zbMATH Keywords: intersection type
zbMATH Keywords: recursive equivalence type
zbMATH Keywords: regressive isols
zbMATH Keywords: co- simple
reviewed by: Leon Harkleroad
MaRDI profile type: Publication
cites work: There exist two regressive sets whose intersection is not regressive
cites work: On the intersection of regressive sets
cites work: The minimum of two regressive isols
cites work: Q5661491
cites work: Models of arithmetic and recursive functions
cites work: Forcing, arithmetic, division rings
cites work: Lower Bounds for Pairs of Recursively Enumerable Degrees
cites work: Realizability and recursive set theory
cites work: Q3926568
cites work: Q3942950
cites work: Q3772202
cites work: Diophantine correct non-standard models in the isols
cites work: Q3819052
description: scientific article; zbMATH DE number 60752