Unboundedness of irreducible decompositions of numerical semigroups

TL;DR

This paper demonstrates that the number of components in irreducible decompositions of certain numerical semigroups can be arbitrarily large, providing a negative answer to a previously posed question.

Contribution

It introduces two families of numerical semigroups with unbounded irreducible decomposition lengths, advancing understanding of their structural complexity.

Findings

Irreducible decomposition lengths are unbounded in these families.

Negative answer to Delgado et al.'s question about boundedness.

Shows the complexity of numerical semigroup decompositions.

Abstract

We present two families of numerical semigroups and show that for each family, the number of required components in an irreducible decomposition cannot be bounded by any given integer. This gives a negative answer to a question raised by Delgado, Garc\'ia-S\'anchez and Rosales.