Weakly half-factorial sets in finite abelian groups

Abstract

We investigate weakly half-factorial sets in finite abelian groups, a concept introduced by J. Śliwa to study half-factorial sets. We fully characterize weakly half-factorial sets in a given group, and determine the maximum cardinality of such a set. This leads to several new results on half-factorial sets; in particular we solve a problem of W. Narkiewicz in some special cases. We also study the arithmetical consequences of weakly-half-factoriality in terms of factorization lengths in block monoids.