On the asymptotic behaviour of the number of distinct factorizations into irreducibles (Q1320532)

A commutative semigroup \(H\) with cancellation and a unit element is called an FF-monoid (finite factorization monoid), provided the number \(f(a)\) of essentially distinct (i.e. up to invertible factors and permutations) factorizations of any \(a\in H\) satisfies \(1\leq f(a)< \infty\). It is proved that if the number of essentially distinct irreducibles dividing some power of \(a\in H\) is finite then \(f(a^ n)= An^{r-1}+ O(n^{r-2})\) holds with a certain \(A>0\) and a suitable explicitly given \(r=r(a)\). This result has interesting applications to the study of the number of distinct factorizations in algebraic number fields.