Longest and Shortest Factorizations in Embedding Dimension Three

Abstract

For a numerical monoid $⟨n_1,\dots ,n_k⟩$ minimally generated by $n_1,\dots ,n_k\in \mathbb{N}$ with $n_1<\cdots <n_k$, the longest and shortest factorization lengths of an element $x$, denoted as $L(x)$ and $\ell (x)$, respectively, follow the identities $L(x+n_1)=L(x)+1$ and $\ell (x+n_k)=\ell (x)+1$ for sufficiently large elements $x$. We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.

References in corpus (1)

• On the set of elasticities in numerical monoids