$\Sigma$-reducibility and $lm$-reducibility of sets and sequences of sets

Limitwise monotonic sets, pairs of sets, and sequences consisting of infinite sets are studied in the paper. The main properties of limitwise monotonic reducibility between two sets, as well as between the set and a pair of sets defined in terms of $\Sigma$-reducibility of the corresponding families of a special form, are considered. In addition, description of $\Sigma$-reducibility of the families of a special form in terms of $lm$-reducibility is obtained. The relationship between the concepts of $lm$-reducibility of the sequences of sets and $\Sigma$-reducibility of the families of a special form for the sequences of sets is demonstrated.