On the class semigroup of a numerical semigroup

Barucci, Valentina; Khouja, Faten

doi:10.1007/s00233-014-9679-8

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…Clearly, for every I ∈ I 0 (S), since 0 ∈ I , there is no other minimal generator of I contained in S. Hence, there exists X ⊆ N \ S such that I = ({0} ∪ X ) + S. In particular, (2) |C (S)| ≤ 2 g , which was already shown in [2] (| • | denotes cardinality). It is also clear that for every g ∈ N \ S, the ideal {0, g } + S ∈ I 0 (S), and so…”

Section: The Ideal Class Monoid Of a Numerical Semigroupmentioning

confidence: 68%

“…The aim of this article is to extend the study of C (S) which was initiated by V. Barucci and F. Khouja in [2]. We prove new estimations of the cardinality of C (S).…”

Section: Introductionmentioning

confidence: 88%

“…Some basic properties of the reduction number of an ideal can be found in [2], and we summarize them in the next result. Proposition 4.12.…”

Section: Lemma 43 Let S Be a Numerical Semigroup With Multiplicity M ...mentioning

confidence: 99%

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Apéry sets and the ideal class monoid of a numerical semigroup

Casabella¹,

D'Anna²,

García-Sánchez³

2023

Preprint

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The aim of this article is to study the ideal class monoid C (S) of a numerical semigroup S introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of C (S). We observe that C (S) is isomorphic to the monoid of ideals of S whose smallest element is 0, which helps to relate C (S) to the Apéry sets and the Kunz coordinates of S. We study some combinatorial and algebraic properties of C (S), including the reduction number of ideals, and the Hasse diagrams of C (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 (C (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 C (S) has at most two quarks.

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Section: The Ideal Class Monoid Of a Numerical Semigroupmentioning

confidence: 68%

“…The aim of this article is to extend the study of C (S) which was initiated by V. Barucci and F. Khouja in [2]. We prove new estimations of the cardinality of C (S).…”

Section: Introductionmentioning

confidence: 88%

See 1 more Smart Citation

Apéry sets and the ideal class monoid of a numerical semigroup

Casabella¹,

D'Anna²,

García-Sánchez³

2023

Preprint

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Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

Casabella,

D’Anna,

García-Sánchez

2023

Mediterr. J. Math.

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The aim of this article is to study the ideal class monoid $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) of a numerical semigroup S introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) . We observe that $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) is isomorphic to the monoid of ideals of S whose smallest element is 0, which helps to relate $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) to the Apéry sets and the Kunz coordinates of S. We study some combinatorial and algebraic properties of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) , including the reduction number of ideals, and the Hasse diagrams of $${\mathscr {C}}\ell (S)$$ C ℓ ( 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 $$({\mathscr {C}}\ell (S),+)$$ ( C ℓ ( 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 $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) has at most two quarks.

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The isomorphism problem for ideal class monoids of numerical semigroups

García-Sánchez

2024

Semigroup Forum

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From any poset isomorphic to the poset of gaps of a numerical semigroup S with the order induced by S, one can recover S. As an application, we prove that two different numerical semigroups cannot have isomorphic posets (with respect to set inclusion) of ideals whose minimum is zero. We also show that given two numerical semigroups S and T, if their ideal class monoids are isomorphic, then S must be equal to T.

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On the class semigroup of a numerical semigroup

Cited by 4 publications

References 5 publications

Apéry sets and the ideal class monoid of a numerical semigroup

Apéry sets and the ideal class monoid of a numerical semigroup

Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

The isomorphism problem for ideal class monoids of numerical semigroups

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