Papers › Apéry sets and the ideal class monoid of a numerical semigroup
Apéry sets and the ideal class monoid of a numerical semigroup
Laura Casabella, Marco D'Anna, Pedro A. García-Sánchez
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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 Ap\'ery 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. Lastly, 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.
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