On large half-factorial sets in elementaryp-groups: Maximal cardinality and structural characterization

Half-factoriality is a central concept in the theory of non-unique factorization, with applications for instance in algebraic number theory. A subsetG ₀ of an abelian group is called half-factorial if the block monoid overG ₀, which is the monoid of all zero-sum sequences of elements ofG ₀, is a half-factorial monoid. In this paper we study half-factorial sets with large cardinality in elementaryp-groups. First, we determine the maximal cardinality of such half-factorial sets, and generalize a result which has been only known for groups of even rank. Second, we characterize the structure of all half-factorial sets with large cardinality (in a sense made precise in the paper). Both results have a direct application in the study of some counting functions related to factorization properties of algebraic integers. This work was supported by the Austrian Science Fund FWF (Project P16770-N12) and by the Austrian-French Program ‘Amadeus 2003–2004’.