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Lengths of irreducible decompositions of numerical semigroups

Pedro Garcia-Sanchez, Christopher O'Neill

TL;DR

The paper investigates how a numerical semigroup can be decomposed into irreducible numerical semigroups and, in particular, the set of possible decomposition lengths. It develops tools based on Apéry sets and special gaps to analyze these decompositions and proves that the set of decomposition lengths forms an interval for multiplicity up to six, with precise structural lemmas and constructive arguments. It further studies ordinary numerical semigroups, constructing irreducible decompositions via $T(F)$ and $I(F)$ that yield a wide range of lengths, and shows that the number of distinct lengths grows with the size of the special-gap set, while the minimum length is unbounded as the multiplicity increases. These results deepen understanding of factorization-like invariants under the non-cancellative intersection operation and reveal both general constraints and flexible constructions within this combinatorial setting.

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.

Lengths of irreducible decompositions of numerical semigroups

TL;DR

The paper investigates how a numerical semigroup can be decomposed into irreducible numerical semigroups and, in particular, the set of possible decomposition lengths. It develops tools based on Apéry sets and special gaps to analyze these decompositions and proves that the set of decomposition lengths forms an interval for multiplicity up to six, with precise structural lemmas and constructive arguments. It further studies ordinary numerical semigroups, constructing irreducible decompositions via and that yield a wide range of lengths, and shows that the number of distinct lengths grows with the size of the special-gap set, while the minimum length is unbounded as the multiplicity increases. These results deepen understanding of factorization-like invariants under the non-cancellative intersection operation and reveal both general constraints and flexible constructions within this combinatorial setting.

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.
Paper Structure (4 sections, 8 theorems, 35 equations, 1 table)

This paper contains 4 sections, 8 theorems, 35 equations, 1 table.

Table of Contents

  1. Introduction
  2. Apéry sets and special gaps
  3. Sets of decomposition lengths in low multiplicity
  4. Ordinary numerical semigroups

Key Result

Lemma 2.1

Let $S$ be a numerical semigroup, and fix a nonzero $m \in S$. Suppose $\mathop{\mathrm{Ap}}\nolimits(S;m) = \{0, a_1, \ldots, a_{m-1}\}$ with $a_i \equiv i \bmod m$ for each $i$, and let $a_k = \max \mathop{\mathrm{Ap}}\nolimits(S;m)$.

Theorems & Definitions (22)

  • Conjecture 1.1
  • Lemma 2.1: numerical
  • Lemma 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 12 more