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

We consider three strong reducibilities, s_1, s_2, Q_1. 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 that s_1 is properly included in s_2, and s_2 is properly included in 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 Δ^0_2 s_2-degrees and minimal Δ^0_2 s_1-degrees. On the other hand, both the Π^0_1 s_2-degrees and the Π^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 Δ^0_2 Q_1-degrees, but the c.e. Q_1-degrees are downwards dense.