Star operations on numerical semigroups: antichains and explicit results

TL;DR

This paper introduces a new order on ideals of numerical semigroups, links it to star operations, and uses it to estimate and explicitly determine the number of star operations, including asymptotic behavior and specific cases.

Contribution

It defines an order on ideals, connects it to star operations, and provides estimates and explicit counts for the number of star operations on numerical semigroups.

Findings

Linked antichains with star operations on semigroups

Provided asymptotic estimates for the number of semigroups with limited star operations

Explicitly determined semigroups with exactly 10 star operations

Abstract

We introduce an order on the set of non-divisorial ideals of a numerical semigroup $S$, and link antichains of this order with the star operations on $S$; subsequently, we use this order to find estimates on the number of star operations on $S$. We then use them to find an asymptotic estimate on the number of nonsymmetric numerical semigroups with $n$ or less star operations, and to determine these semigroups explicitly when $n=10$.