A hierarchy for the plus cupping turing degrees

We say that a computably enumerable (c. e.) 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 PC_n 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 [14] showing that LC₁ ⊂ LC₂, as well as extending the Harrington plus-cupping theorem [8].