The number of addends in the decomposition of an element of a numerical semigroup into atoms

TL;DR

This paper demonstrates that for any small set of integers greater than one, there exists a numerical semigroup and element with decompositions into atoms matching that set, and proposes related conjectures.

Contribution

It establishes the existence of numerical semigroups with elements whose atom decomposition counts match any small set of integers, and introduces three conjectures.

Findings

Existence of such semigroups for sets with up to three elements

Construction methods for elements with prescribed atom decomposition counts

Three conjectures related to atom decompositions in numerical semigroups

Abstract

We prove that for every nonempty set $\Sigma$ of integers bigger than $1$, which has at most three elements, there exists a numerical semigroup $T$ and an element $x$ of $T$ such that a natural number $n$ is the number of atoms in a decomposition of $x$ into atoms if and only if $n$ belongs to $\Sigma$. We also propose three related conjectures.