Length density and numerical semigroups

TL;DR

This paper investigates the concept of length density, a factorization invariant, within numerical semigroups, providing insights into how factorization lengths deviate from being continuous intervals.

Contribution

It introduces and analyzes the length density invariant specifically for numerical semigroups, expanding understanding of factorization properties in these structures.

Findings

Length density quantifies deviation from interval structure in factorizations.

Numerical semigroups exhibit diverse length density behaviors.

The study offers new tools for analyzing factorization in additive semigroups.

Abstract

Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative integers).