The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

Let H be an atomic monoid. For x is an element of H, let L(x) denote the set of all possible lengths of factorizations of x into irreducibles. The system of sets of lengths of H is the set L( H) = {L(x) vertical bar x is an element of H}. On the other hand, the elasticity of x, denoted by rho(x), is the quotient sup L(x)/inf L(x) and the elasticity of H is the supremum of the set {rho(x) vertical bar x is an element of H}. The system of sets of lengths and the elasticity of H both measure how far H is from being half-factorial, i.e. vertical bar L(x)vertical bar = 1 for each x is an element of H. Let C denote the collection comprising all submonoids of finite-rank free commutative monoids, and let C-d = {H is an element of C vertical bar rank(H) = d}. In this paper, we study the system of sets of lengths and the elasticity of monoids in C. First, we construct for each d >= 2 a monoid in C-d having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in C-1. Here we use our construction to extend this result to C-d for any d >= 2. On the other hand, it has been recently conjectured that the elasticity of any monoid in C is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in C-2 and for any monoid in C whose corresponding convex cone is polyhedral.