Some problems related to the ideal class monoid of a numerical semigroup

Abstract: Let $S$ be a numerical semigroup (a submonoid of the set of non-negative integers under addition such that $\max(\mathbb{Z}\setminus S)$ exists). A non-empty set of integers $I$ is said to be an ideal of $S$ if $I+S\subseteq I$ and $I$ has a minimum. If $I$ and $J$ are ideals of $S$, we write $I\sim J$ if there exists an integer $z$ such that $I=z+J$. The ideal class monoid of $S$ is defined as the set of ideals of $S$ modulo this relation, where addition of two classes $[I]$ and $[J]$ is defined as $[I]+[J]=[I+J]$, with $I+J=\{i+j\mid i\in I, j\in J\}$.

An ideal $I$ is said to be normalized if $\min(I)=0$. The set of normalized ideals of $S$, denoted by $\mathfrak{I}_0(S)$, is a monoid isomorphic to the ideal class monoid of $S$ [1].

It is known that if $S$ and $T$ are numericals semigroups for which $\mathfrak{I}_0(S)$ is isomorphic to $\mathfrak{I}_0(T)$, then $S$ and $T$ must be the same numerical semigroup [2].

On $\mathfrak{I}_0(S)$ we can define a partial order $\preceq$ as $I\preceq J$ if there exists $K\in \mathfrak{I}_0(S)$ such that $I+K=J$. We know that if $S$ and $T$ are numerical semigroups with multiplicity three such that the poset $(\mathfrak{I}_0(S),\preceq)$ is isomorphic to the poset $(\mathfrak{I}_0(T),\preceq)$, then $S$ and $T$ are the same numerical semigroup [3]. However, if we remove the multiplicity three condition, this poset isomorphsm problem is still open.

In [4], we study the case when the poset $(\mathfrak{I}_0(S),\preceq)$ is a lattice. We show that this is the case if and only if the multiplicity of $S$ is at most four.

References:

1. L. Casabella, M. D'Anna, P. A. García-Sánchez, Apéry sets and the ideal class monoid of a numerical semigroup, Mediterr. J. Math. 21, 7 (2024).

2. P. A. García-Sánchez, The isomorphism problem for ideal class monoids of numerical semigroups, Semigroup Forum 108 (2024), 365--376.

3. S. Bonzio, P. A. García-Sánchez, The poset of normalized ideals of numerical semigroups with multiplicity three, to appear in Comm. Algebra.

4. S. Bonzio, P. A. García-Sánchez, When the poset of the ideal class monoid of a numerical semigroup is a lattice, arXiv:2412.07281.

Mathematics

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2025)

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