Discontinuity of Cappings in the Recursively Enumerable Degrees and Strongly Nonbranching Degrees

We construct an r. e. degree a which possesses a greatest a‐minimal pair b 0 , b 1 , i.e., r. e. degrees b 0 and b 1 such that b 0 , b 1 < a, b 0 ∩ b 1 = a, and, for any other pair c 0 , c 1 with these properties, c 0 ≤ b i and c 1 ≤ b 1‐ i for some i ≤ 1. By extending this result, we show that there are strongly nonbranching degrees which are not strongly noncappable. Finally, by introducing a new genericity concept for r. e. sets, we prove a jump theorem for the strongly nonbranching and strongly noncappable r. e. degrees. Mathematics Subject Classification : 03D25.