Good Semigroups Research Articles

Almost symmetric good semigroups

The class of good semigroups is a class of subsemigroups of Nh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}^h$$\\end{document}, 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 Nh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}^h$$\\end{document}, 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.

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

In this article, we discuss some applications of the construction of the Apéry set of a good semigroup in mathbb {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.

Partition of the complement of good semigroup ideals and Apéry sets

Good semigroups form a class of submonoids of containing the value semigroups of curve singularities. In this article, we describe a partition of the complement of good semigroup ideals. As main application, we describe the Apéry sets of good semigroups with respect to arbitrary elements. This generalizes to any the results in the study of D’Anna et al., which are proved in the case d = 2 and only for the standard Apéry set with respect to the smallest nonzero element. Several new results describing good semigroups in are also provided.

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

Good subsemigroups of {mathbb {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. For 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, 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.

The Type of a Good Semigroup and the Almost Symmetric Condition

We study the type and the almost symmetric condition for good subsemigoups of $${\mathbb {N}}^2$$, a class of semigroups containing the value semigroups of curve singularities with two branches. We define the type in term of a partition of a specific set associated to the semigroup and we show that this definition generalizes the well-known notion of type of a numerical semigroup and has a good behaviour with respect to the corresponding concept for algebroid curves. Then, we study almost symmetric good semigroups, their connections with maximal embedding dimension good semigroups and their Apery set, generalizing to this context several existent known results.

Arf good semigroups with fixed genus

The concept of Arf numerical semigroup plays a significant role in the study of the equivalence between algebroid branches. Namely, two algebroid branches are said to be equivalent if their Arf closures have the same value semigroup, that is an Arf numerical semigroup. By introducing the concept of Arf good semigroups of $$\mathbb {N}^r$$, it is possible to extend the aforementioned equivalence to the more general context of algebroid curves with $$r>1$$ branches. Arf good semigroups can be completely described by their multiplicity trees that are combinatorial objects whose study is independent from the ring theory context. In this paper we give an algorithm for the computation of all Arf numerical semigroups with a given genus. Moreover, we generalize the concept of genus of a numerical semigroup to good semigroups of $$\mathbb {N}^r$$ and we give a procedure to calculate all Arf good semigroups of $$\mathbb {N}^r$$ with a given genus.

Duality on value semigroups

We establish a combinatorial counterpart of the Cohen-Macaulay duality on a class of curve singularities which includes algebroid curves. For such singularities the value semigroup and the value semigroup ideals of all fractional ideals satisfy axioms that define so-called good semigroups and good semigroup ideals. We prove that each good semigroup admits a canonical good semigroup ideal which gives rise to a duality on good semigroup ideals. We show that the Cohen-Macaulay duality and our good semigroup duality are compatible under taking values.

Arf good semigroups

In this paper, we study the property of the Arf good subsemigroups of [Formula: see text], with [Formula: see text]. We give a way to compute all the Arf semigroups with a given collection of multiplicity branches. We also deal with the problem of determining the Arf closure of a set of vectors and of a good semigroup, extending the concept of characters of an Arf numerical semigroup to Arf good semigroups.

Arf good semigroups with fixed conductor

In this paper we present a new algorithm to calculate the set of Arf numerical semigroups with given conductor. Moreover, we extend the previous problem to the class of good semigroups, presenting procedures to compute the set of Arf good semigroups of $$\mathbb {N}^r$$ with a fixed conductor $$\mathbf c \in \mathbb {N}^r$$ , for all $$r \in \mathbb {N}$$ .

Good subsemigroups of ℕn

Value semigroups of non-irreducible singular algebraic curves and their fractional ideals are submonoids of [Formula: see text] that are closed under infimums, have a conductor and fulfill a special compatibility property on their elements. Monoids of [Formula: see text] fulfilling these three conditions are known in the literature as good semigroups and there are examples of good semigroups that are not realizable as the value semigroup of an algebraic curve. In this paper, we consider good semigroups independently from their algebraic counterpart, in a purely combinatorial setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique. Moreover, we give a constructive way to compute the canonical ideal and the Arf closure of a good subsemigroup when [Formula: see text].

Good B-quasi-Ehresmann semigroups

A semigroup S with a sub-band B is called a good B-quasi-Ehresmann semigroup if it is a B-semiabundant semigroup satisfying the congruence condition such that aB(a*)B(b+)b+ ⊆ B((ab)+)abB((ab)*) for all a, b ∈ S. We show that every good B-quasi-Ehresmann semigroup has a global representation and a standard representation. As a special case, the structure of good quasi-adequate semigroups is described.