Structural properties of limitwise monotonic reducibility of sequences of sets

The paper is devoted to the study of limitwise monotonic sets, as well as to the investigation of the main structural properties of limitwise monotonic reducibility ($lm$-reducibility) between set and sequence of sets. Limitwise monotonic reducibility can be regarded as a special case of $\Sigma$-reducibility defined on families of initial segments of natural numbers. In this paper, $lm$-reducibility between set and sequence consisting of infinite sets has been considered in the language of the limitwise monotonic operator. The main result of this paper is the proof of the absence of the least non-limitwise monotonic sequence with respect to $lm$-reducibility between set and sequence of sets. This result as been proved with the help of the infinite injury priority method with the use of the tree of strategies. The result presented in this paper is a generalization of the result that there are no least non-limitwise monotonic $\Sigma^0_2$-set under $lm$-reducibility of sets.