On limitwise monotonic reducibility of $\Sigma_2^0$-sets

In this paper, we study the properties of lm-reducibility of sets belonging to the class of $\Sigma_2^0$-sets. In particular, we prove the existence of incomparable $\Sigma_2^0$-sets with respect to lm-reducibility. In addition, we construct an infinite uniform sequence of incomparable $\Sigma_2^0$-sets relative to lm-reducibility and show that every countable partial order can be embedded into the class of all lm-degrees of $\Sigma_2^0$-sets.