Minimal degrees and downwards density in some strong positive reducibilities and quasi-reducibilities

Abstract

Abstract We consider three strong reducibilities, $s_{1}, s_{2}, Q_{1}$ (where we identify a reducibility $\leqslant _r$ with its index $r$). The first two reducibilities can be viewed as injective versions of $s$-reducibility, whereas $Q_1$-reducibility can be viewed as an injective version of $Q$-reducibility. We have, with proper inclusions, $s_{1} \subset s_{2} \subset s$. It is well known that there is no minimal $s$-degree, and there is no minimal $Q$-degree. We show on the contrary that there exist minimal $\varDelta ^{0}_{2}$$s_{2}$-degrees and minimal $\varDelta ^{0}_{2}$$s_{1}$-degrees. On the other hand, both the $\varPi ^{0}_{1}$$s_{2}$-degrees and the $\varPi ^{0}_{1}$$s_{1}$-degrees are downwards dense. By the isomorphism of the $s_1$-degrees with the $Q_1$-degrees induced by complementation of sets, it follows that there exist minimal $\varDelta ^0_2$$Q_1$-degrees, but the c.e. $Q_{1}$-degrees are downwards dense.