Extending the cooper minimal pair theorem

In the study of cappable and noncappable properties of the recursively enumerable (r.e.) degrees, Lempp suggested a conjecture which asserts that for all. r.e. degreesa andb, ifa≰b then there exists an r.e. degreec such thatc≤a andc≰b andc is cappable. We shall prove in this paper that this conjecture holds under the condition thata is high. Working below a high r.e. degreeh, we show that for any r.e. degreeb withh≰b, there exist r.e. degreesa ₀ anda ₁ such thata ₀,a ₁≰b,a ₀,a ₁≰h, anda ₀ anda ₁ form a minimal pair. Tais research is supported by the National Natural Science Foundation of China (No. 19971090). ZHANG Zaiyue was born in 1961. He received the Ph.D. degree in mathematics from the Institute of Software, the Chinese Academy of Sciences in 1995. He is an associate professor of the Department of Computer Science, Industry College, Yangzhou University. His current research interests include recursion theory, complexity of computation and set theory.