On the number of factorizations of an element in an atomic monoid

Abstract

Let S be a reduced commutative cancellative atomic monoid. If s is a nonzero element of S, then we explore problems related to the computation of η(s), which represents the number of distinct irreducible factorizations of s∈S. In particular, if S is a saturated submonoid of Nd, then we provide an algorithm for computing the positive integer r(s) for which 0<limn→∞η(sn)nr(s)−1<∞. We further show that r(s) is constant on the Archimedean components of S. We apply the algorithm to show how to compute limn→∞η(sn)nr(s)−1 and also consider various stability conditions studied earlier for Krull monoids with finite divisor class group.