Extremal factorization lengths of elements in commutative, cancellative semigroups

Abstract

For a numerical semigroup $S := \langle n_1, \dots, n_k \rangle$ with minimal generators $n_1 < \cdots < n_k$, Barron, O'Neill, and Pelayo showed that $L(s+n_1) = L(s) + 1$ and $\ell(s+n_k) = \ell(s) + 1$ for all sufficiently large $s \in S$, where $L(s)$ and $\ell(s)$ are the longest and shortest factorization lengths of $s \in S$, respectively. For some numerical semigroups, $L(s+n_1) = L(s) + 1$ for all $s \in S$ or $\ell(s+n_k) = \ell(s) + 1$ for all $s \in S$. In a general commutative, cancellative semigroup $S$, it is also possible to have $L(s+m) = L(s) + 1$ for some atom $m$ and all $s \in S$ or to have $\ell(s+m) = \ell(s) + 1$ for some atom $m$ and all $s \in S$. We determine necessary and sufficient conditions for these two phenomena. We then generalize the notions of Kunz posets and Kunz polytopes. Each integer point on a Kunz polytope corresponds to a commutative, cancellative semigroup. We determine which integer points on a given Kunz polytope correspond to semigroup in which $L(s+m) = L(s) + 1$ for all $s$ and similarly which integer points yield semigroups for which $\ell(s+m) = \ell(s) + 1$ for all $s$.