Limitwise monotonic reducibility on sets and on pairs of sets

We study limitwise monotonic sets and pairs of sets. We investigate the properties of limitwise monotonic reducibility between sets and pairs of sets defined in terms of $\Sigma$-reducibility corresponding to initial segment of sets. In addition, we obtain a description of $\Sigma$-reducibility of families of a special form in terms of $lm$-reducibility. At the same time we show the relationship of concepts of $lm$-reducibility and $\Sigma$-reducibility between the pairs of sets.