Extremal factorization lengths of elements in commutative, cancellative semigroups

TL;DR

This paper characterizes when extremal factorization length properties hold in general commutative, cancellative semigroups, extending known results from numerical semigroups and connecting to geometric objects called Kunz polytopes.

Contribution

It provides necessary and sufficient conditions for extremal factorization length phenomena and generalizes Kunz concepts to broader semigroup classes.

Findings

Characterization of when $L(s+m) = L(s) + 1$ for all $s$

Characterization of when $ ext{ell}(s+m) = ext{ell}(s) + 1$ for all $s$

Identification of Kunz polytope points corresponding to these properties

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 $s$.