A note on sets of lengths of powers of elements of finitely generated monoids

Let H be a finitely generated commutative, cancellative monoid. Then every nonunit a is an element of H decomposes (in general in a highly nonunique way) into a product a = u(1) ..... u(n) (19.1) of irreducible elements (atoms) of H. The integer n is called the length of the factorization (19.1). We call L(a) = {n is an element of N vertical bar a has a factorization into n irreducible elements of H} the set of lengths of a. It is known that sets of lengths of elements of H have the following structure: they are, up to bounded initial and final segments, a union of arithmetical progressions with bounded distance. We say that two sets of lengths are of the same type if their initial and final segments coincide (up to a shift) and if their central parts have the same period. Clearly, the set of equivalence classes with respect to this equivalence relation is finite. In the present note we examine the structure of sets of lengths of powers of elements of H. We prove a theorem which asserts that there exist constants N, B G N (which only depend on H) such that the following holds: if a is an element of H and k, l >= B with k equivalent to l mod N, then L(a(k)) and L(a(l)) are of the same type. This result carries over to Krull monoids with finite divisor class group, e.g. rings of integers of algebraic number fields. In a weaker form it is true for every Krull monoid.