On 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 $x \in H$, let $\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\mathcal{L}(H) = \{\mathsf{L}(x) \mid x \in H\}$. On the other hand, the elasticity of $x$, denoted by $\rho(x)$, is the quotient $\sup \mathsf{L}(x)/\inf \mathsf{L}(x)$ and the elasticity of $H$ is the supremum of the set $\{\rho(x) \mid x \in H\}$. The system of sets of lengths and the elasticity of $H$ both measure how far is $H$ from being half-factorial, i.e., $|\mathsf{L}(x)| = 1$ for each $x \in H$. Let $\mathcal{C}$ denote the collection comprising all submonoids of finite-rank free commutative monoids, and let $\mathcal{C}_d = \{H \in \mathcal{C} \mid \text{rank}(H) = d\}$. In this paper, we study the system of sets of lengths and the elasticity of monoids in $\mathcal{C}$. First, we construct for each $d \ge 2$ a monoid in $\mathcal{C}_d$ 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 $\mathcal{C}_1$. Here we use our construction to extend this result to $\mathcal{C}_d$ for any $d \ge 2$. On the other hand, it has been recently conjectured that the elasticity of any monoid in $\mathcal{C}$ is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in $\mathcal{C}_2$ and for any monoid in $\mathcal{C}$ whose corresponding convex cone is polyhedral.