On $tt$-degrees of recursively enumerable Turing degrees

The main result of the paper asserts that if $A$ is a semirecursive $\eta$-hyperhypersimple set, then for every set $B$ with $A\equiv_TB$ there exists a recursive set $C$ such that $C\leq_mA$ and $C\leqslant_{tt}B$. If $B$ is recursively enumerable, then $C\leqslant_qB$. A corollary asserts that if a $tt$-degree contains an $\eta$-maximal semirecursive set, then it is a minimal element in the semilattice of all $tt$-degrees.
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