Factorizations into irreducible numerical semigroups

TL;DR

This paper investigates how numerical semigroups can be factored into irreducible components, revealing that the sets of lengths of such factorizations cover all integers greater than or equal to 2.

Contribution

It proves that the unions of sets of lengths of factorizations into irreducible numerical semigroups are all equal to the set of integers greater than or equal to 2.

Findings

Sets of lengths of factorizations cover all integers ≥ 2

Every numerical semigroup can be decomposed into irreducible ones

Unions of length sets are identical across factorizations

Abstract

Every numerical semigroup can be expressed as an intersection of irreducible numerical semigroups. We show that the unions of sets of lengths of factorizations of numerical semigroups into irreducible numerical semigroups are all equal to $\mathbb{N}_{\ge 2}$.