$\hat{I}»$-fold near-factorizations of groups

Abstract

We initiate the study of $\hat{I}»$-fold near-factorizations of groups with $\hat{I}»>1$. While $\hat{I}»$-fold near-factorizations of groups with $\hat{I}»=1$ have been studied in numerous papers, this is the first detailed treatment for $\hat{I}»>1$. We establish fundamental properties of $\hat{I}»$-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of $\hat{I}»$-fold near-factorizations, including upper bounds on $\hat{I}»$. We present three constructions of infinite families of $\hat{I}»$-fold near-factorizations, highlighting the characterization of two subfamilies of $\hat{I}»$-fold near-factorizations. We discuss a computational approach to $\hat{I}»$-fold near-factorizations and tabulate computational results for abelian groups of small order.