A study of factorizations of elements in monoids via related graphical structures

Abstract

Recently, there has been a flourish of research related to graphically representing factorizations of elements in integral domains and, more generally, in commutative cancellative monoids. We recall the notions of irreducible divisor graphs and simplicial complexes introduced over the past decade and introduce a new related structure, the compressed irreducible divisor simplicial complex. We compare this structure with previous constructions and extend previous results using these structures to give information about factorizations of given elements. We conclude by studying unique and non-unique factorization. Specifically, we give a characterization of prime elements using these graphical structures and improve known bounds on elasticities of elements.