Lengths of irreducible decompositions of numerical semigroups

TL;DR

This paper investigates the set of all possible lengths of irreducible decompositions of numerical semigroups, proving a conjecture for certain cases and constructing examples with large intervals of decomposition lengths.

Contribution

It proves the conjecture that the set of decomposition lengths forms an interval for semigroups with smallest positive element at most six and constructs families with arbitrarily large decomposition length intervals.

Findings

The set of decomposition lengths is always an interval for semigroups with smallest positive element ≤ 6.

Constructed families of semigroups with arbitrarily large maximum decomposition lengths.

Proved the conjecture for specific classes of numerical semigroups.

Abstract

A numerical semigroup is an additive subsemigroup of the natural numbers that contains zero and has finite complement. A numerical semigroup is irreducible if it cannot be written as an intersection of numerical semigroups properly containing it. It is known that every numerical semigroup can be decomposed as an intersection of irreducible numerical semigroups, but there can be multiple such decompositions, even when irredundancy is required. In this paper, we study the set of all decomposition lengths of a given numerical semigroup. It is conjectured that the set of decomposition lengths is always an interval; we prove this conjecture for numerical semigroups whose smallest positive element is at most six. Additionally, we examine a class of numerical semigroups that was recently shown to achieve arbitrarily large minimum decomposition length, and construct a family of irreducible decompositions whose lengths form a large interval.