Immunity properties of the s-degrees

We investigate immunity properties of the s-degrees. In particular we show that neither the immune nor the hyperimmune s-degrees are upwards closed since there exist $\Delta^0_2$ s-degrees $a \leq_s b$ such that $a$ is hyperimmune, but $b$ is immune free. We also show that there is no hyperhyperimmune $\Pi^0_2$ set A such that $\overline{K} \leq_{\overline{s}} A$, where $\overline{K}$ is the complement of the halting set and $ \leq_{\overline{s}}$ denotes the finite-branch version of s-reducibility.