Complementary Numerical Sets

TL;DR

This paper explores the relationship between numerical sets and Young diagrams, defining a complement operation and analyzing properties related to closure under addition, with implications for numerical semigroups.

Contribution

It introduces a novel complement operation on numerical sets via Young diagrams and investigates its properties, especially regarding closure under addition.

Findings

The complement of a numerical set corresponds to a specific Young diagram transformation.

Conditions for closure under addition are characterized for both the original and complemented sets.

The work links numerical sets to Young diagrams, enriching the structural understanding of numerical semigroups.

Abstract

A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements. We determine various properties of $\widetilde{S}$, particularly with an eye to closure under addition (for both $S$ and $\widetilde{S}$), which promotes a numerical set to become a numerical semigroup.