Limitwise monotonic reducibility of sets and $\Sigma$-definability of abelian groups

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 (for short we will also write $lm$-reducibility) between sets. In this paper, we obtain a description of the algorithmic dependence between the limitwise monotonic reducibility of sets, which defined in terms of the $\Sigma$-reducibility of the families of initial segments, and the $\Sigma$-definability of abelian groups.