The Type of a Good Semigroup and the Almost Symmetric Condition

D'Anna, Marco; Guerrieri, Lorenzo; Micale, Vincenzo

doi:10.1007/s00009-019-1467-y

Cited by 7 publications

(36 citation statements)

References 16 publications

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“…The proof of the second formula is analogous to the first one. (6,10,10,18), (7,3,2,10), (7,6,4,18), (7,7,6,18), (7,7,8,18), (7,7,9,18), (7,9,6,18), (7,10,8,18), (7,10,9,18), (8,3,2,10), (8,6,4,18), (8,7,6,18), (8,7,8,18), (8,7,…”

Section: Definition 21mentioning

confidence: 99%

“…In this subsection, we want to relate the genus and the type of a good semigroup S ⊆ N 2 by generalizing a well known inequality that holds in the case of numerical semigroups. First of all, we recall the concept of the type of a good semigroup by following the definition introduced in [8] that extends the one initially given in [1].…”

Section: On the Type Of A Good Semigroupmentioning

confidence: 99%

“…Given a good semigroup S ⊆ N 2 , let us consider a set A ⊆ S such that there exists c ∈ N 2 with c + N 2 ⊆ S\A. As described in [8], it is possible to build up a partition of such a set A, in the following way.…”

Section: On the Type Of A Good Semigroupmentioning

confidence: 99%

“…In [15], it was proved the inequality c(S) ≤ (t(S)+1)l(S) for numerical semigroups. The type of a good semigroup was originally defined in [1] for the good semigroups such that S − M is a relative good ideal of S and it was recently generalized in [8]. We conclude the paper asking if this inequality holds for good semigroups with respect to the generalized version of the definition of the type.…”

Section: Introductionmentioning

confidence: 97%

“…Thus, such semigroups can be seen as a natural generalization of numerical semigroups and can be studied using a more combinatorial approach without necessarily referring to the ring theory context. In recent works [8,9,19], some notable elements and properties of numerical semigroups have been generalized to the case of good semigroups.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture

Maugeri

Zito

2020

Mediterr. J. Math.

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Good subsemigroups of $${\mathbb {N}}^d$$ N d have been introduced as the most natural generalization of numerical ones. Although their definition arises by taking into account the properties of value semigroups of analytically unramified rings (for instance the local rings of an algebraic curve), not all good semigroups can be obtained as value semigroups, implying that they can be studied as pure combinatorial objects. In this work, we are going to introduce the definition of length and genus for good semigroups in $${\mathbb {N}}^d$$ N d . For $$d=2$$ d = 2 , we show how to count all the local good semigroups with a fixed genus through the introduction of the tree of local good subsemigroups of $${\mathbb {N}}^2$$ N 2 , generalizing the analogous concept introduced in the numerical case. Furthermore, we study the relationships between these elements and others previously defined in the case of good semigroups with two branches, as the type and the embedding dimension. Finally, we show that an analogue of Wilf’s conjecture fails for good semigroups in $${\mathbb {N}}^2$$ N 2 .

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Section: Definition 21mentioning

confidence: 99%

Section: On the Type Of A Good Semigroupmentioning

confidence: 99%

Section: On the Type Of A Good Semigroupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture

Maugeri

Zito

2020

Mediterr. J. Math.

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Properties and applications of the Apéry set of good semigroups in $$\mathbb {N}^d$$

Guerrieri

Maugeri²,

Micale³

2022

J Algebr Comb

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In this article, we discuss some applications of the construction of the Apéry set of a good semigroup in $$\mathbb {N}^d$$ N d given in (Commun Algebra 49(10):4136–4158, 2021). In particular, we study the duality of a symmetric and almost symmetric good semigroup, the Apéry set of non-local good semigroups and the Apéry set of value semigroups of plane curves.

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Almost symmetric good semigroups

Casabella

D'Anna

2023

Ricerche mat

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The class of good semigroups is a class of subsemigroups of $${\mathbb {N}}^h$$ N h , that includes the value semigroups of rings associated to curve singularities and their blowups, and allows to study combinatorically the properties of these rings. In this paper we give a characterization of almost symmetric good subsemigroups of $${\mathbb {N}}^h$$ N h , extending known results in numerical semigroup theory and in one-dimensional ring theory, and we apply these results to obtain new results on almost Gorenstein one-dimensional analytically unramified rings.

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The Type of a Good Semigroup and the Almost Symmetric Condition

Cited by 7 publications

References 16 publications

The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture

The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture

Properties and applications of the Apéry set of good semigroups in $$\mathbb {N}^d$$

Almost symmetric good semigroups

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