On the asymptotic behavior of unions of sets of lengths in atomic monoids

Abstract

Let M be a commutative cancellative atomic monoid. We use unions of sets of lengths in M to construct the V-Delta set of M . We first derive some basic properties of V-Delta sets and then show how they offer a method to investigate the asymptotic behavior of the sizes of unions of sets of lengths. A central focus of number theory is the study of number theoretic functions and their asymptotic behavior. This has led to similar investigations concerning non-unique factorizations in integral domains and moniods. Suppose that M is a commutative cancellative monoid in which each nonunit can be factored into a product of irreducible elements (such a monoid is known as atomic). For a nonunit x in M , let L(x) represent the maximum length of a factorization of x into irreducibles and l(x) the minimum such length. The functions L(x) = lim k→∞ L(xn) n and l(x) = lim k→∞ l(xn)