Longest and Shortest Factorizations in Embedding Dimension Three

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 $\mathrm{\ell }(x)$, respectively, follow the identities $L(x+n_1)=L(x)+1$ and $\mathrm{\ell }(x+n_k)=\mathrm{\ell }(x)+1$ for sufficiently large elements $x$. We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.