Embedding Dimension of a Good Semigroup

Maugeri, Nicola; Zito, Giuseppe

doi:10.1007/978-3-030-40822-0_13

Cited by 8 publications

(22 citation statements)

References 10 publications

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“…Now we want to introduce a useful formula for the computation of the genus. As in [6] and in [19], we will work on the equivalent structure of semiring Γ S to simplify the notation.…”

Section: Definition 21mentioning

confidence: 99%

“…We denote by A f (S) the set of finite maximals in S and by A(S) := A f (S) ∪ S ∞ the set of all maximals in S. Furthermore, we call I A (S) the set of all irreducible maximals in S. It is easy to see that the set of irreducible maximals is finite and in [19] it is proved that S is generated by these elements as a semiring.…”

Section: Definition 21mentioning

confidence: 99%

“…In this section we want to repeat the same process with the good semigroups, building a tree of local good semigroups of N 2 , where the gth level of the tree collects all local good semigroups with genus g. To follow this idea, we will show that, in this case, the analogous of minimal generators are the tracks of the good semigroup originally defined in [19].…”

Section: The Tree Of Good Semigroups Of N 2 By Genusmentioning

confidence: 99%

“…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%

“…In Sect. 3, we repeat the same idea for good semigroups S ⊆ N 2 ; in this case, the tracks of the good semigroup, defined in [19], will have the role that minimal generators played in case of numerical one. To do this, we prove that every good semigroup of genus g can be obtained removing a track from one of genus g − 1 (Theorem 3.4) and that by removing a track from a good semigroup of genus g we obtain a good semigroup of genus g + 1 (Theorem 3.5).…”

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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“…Now we want to introduce a useful formula for the computation of the genus. As in [6] and in [19], we will work on the equivalent structure of semiring Γ S to simplify the notation.…”

Section: Definition 21mentioning

confidence: 99%

Section: Definition 21mentioning

confidence: 99%

Section: The Tree Of Good Semigroups Of N 2 By Genusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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Embedding Dimension of a Good Semigroup

Cited by 8 publications

References 10 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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