On co-simple isols and their intersection types
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.
scientific article; zbMATH DE number 60752
scientific article