Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

The aim of this article is to study the ideal class monoid Cl(S) of a numerical semigroup S introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of Cl(S). We observe that Cl(S) is isomorphic to the monoid of ideals of S whose smallest element is 0, which helps to relate Cl(S) to the Apéry sets and the Kunz coordinates of S. We study some combinatorial and algebraic properties of Cl(S), including the reduction number of ideals, and the Hasse diagrams of Cl(S) with respect to inclusion and addition. From these diagrams, we can recover some notable invariants of the semigroup. Finally, we prove some results about irreducible elements, atoms, quarks, and primes of (Cl(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 Cl(S) has at most two quarks.