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

We consider three strong reducibilities, s(1), s(2), Q(1) (where we identify a reducibility <=(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 of s(2) subset of 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 Delta(0)(2) s(2) -degrees and minimal Delta(0)(2) s(1)-degrees. On the other hand, both the Pi(0)(1) s(2) -degrees and the Pi(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 Delta(0)(2) Q(1)-degrees, but the c.e. Q(1)-degrees are downwards dense.