On the relative complexity of hard problems for complexity classes without complete problems (Q1112017)

The main result states that any recursive sequence ascending relatively to polynomial time reducibility (p-r-ascending) of recursive sets has no minimal p-r-upper bound. As a corollary it is proved that any complexity class closed under the direct sum possesses either complete problems or no minimal hard problems. Another corollary claims that provided \(P\neq NP\), the partial ordering of NP relative to polynomial Turing reducibility (resp. m-reducibility) is not complete.