Abstract

We say that a computably enumerable degree a is plus-cupping, if for every c. e. degree x with 0 < x≤ a, there is a c. e. degree y ≠ 0’ such that x ∨ y=0’. We say that a is n-plus-cupping, if for every c. e. degree x, if 0 < x ≤ a, then there is a lown c. e. degree l such that x ∨ l=0’. Let PC and PCn be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then PC1 ⊆ PC2⊆ PC3 = PC. In this paper we show that PC1 ⊂ PC2, so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [li-wu-zhang] showing that LC1 ⊂ LC2, as well as extending the Harrington plus-cupping theorem [harrington1978].