Hypersimplicity and Semicomputability in the Weak Truth Table Degrees

Abstract. We study the classes of hypersimple and semicomputable sets as well as their intersection in the weak truth table degrees. We construct degrees that are not bounded by hypersimple degrees outside any non-trivial upper cone of Turing degrees and show that the hypersimple-free c.e. wtt degrees are downwards dense in the c.e. wtt degrees. Moreover, we consider the sets that are both hypersimple and semicomputable, characterize them as the (bi-infinite) c.e. cuts of computable orderings of N of order type ω+ω ∗ and study their wtt degrees. We show that there are hypersimple degrees that are not bounded by any hypersimple semicomputable degree, investigate relationships with the join and look for maximal and minimal elements of related classes. 1