Ap\'ery sets and the ideal class monoid of a numerical semigroup

TL;DR

This paper investigates the ideal class monoid of a numerical semigroup, establishing bounds, exploring its algebraic structure, and relating it to Apéry sets and Kunz coordinates to understand semigroup invariants.

Contribution

It introduces new bounds on the size of the ideal class monoid and relates it to Apéry sets, providing new insights into the algebraic and combinatorial properties of numerical semigroups.

Findings

Bounds on the cardinality of the ideal class monoid

Isomorphism between the monoid and ideals with smallest element 0

Characterization of irreducible semigroups via the monoid structure

Abstract

The aim of this article is to study the ideal class monoid $\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We observe that $\mathcal{C}\ell(S)$ is isomorphic to the monoid of ideals of $S$ whose smallest element is 0, which helps to relate $\mathcal{C}\ell(S)$ to the Ap\'ery sets and the Kunz coordinates of $S$. We study some combinatorial and algebraic properties of $\mathcal{C}\ell(S)$, including the reduction number of ideals, and the Hasse diagrams of $\mathcal{C}\ell(S)$ with respect to inclusion and addition. From these diagrams we can recover some notable invariants of the semigroup. Lastly, we prove some results about irreducible elements, atoms, quarks and primes of $(\mathcal{C}\ell(S),+)$. Idempotent ideals coincide with over-semigroups and idempotent quarks correspond to unitary extensions of the semigroup. We show that a numerical semigroup is irreducible if and only if $\mathcal{C}\ell(S)$ has at most two quarks.