一类由Kunz和Waldi定义的数值半群

交换可消半群中元素的极值分解长度

Extremal factorization lengths of elements in commutative, cancellative semigroups

该研究确定了交换可消半群中原子使元素极值分解长度逐次加1的充要条件，并推广Kunz偏序集与多面体概念，明确对应半群性质的整数点特征。

The tree of numerical semigroups with low multiplicity

We show that the number of numerical semigroups with multiplicity three, four or five and fixed genus is increasing as a function in the genus. To this end we use the Kunz polytope for these multiplicities. Counting numerical semigroups with fixed multiplicity and genus is then an integer partition problem with some extra conditions (those of membership to the Kunz polytope). For the particular case of multiplicity four, we are able to prove that the number of numerical semigroups with multiplicity four and genus $g$ is the number of partitions $x+y+z=g+6$ with $0<x\le y\le z$, $x\neq 1$, $y\n…

关于数值半群极小表示的基数

On the cardinality of minimal presentations of numerical semigroups

本文从组合（偏序集理论）视角，利用Kunz nil半群概念研究数值半群极小表示的基数问题，给出了显式构造与一般界，并提供了避免多面体几何的自包含介绍。

导子模与射影单项式曲线坐标环的 Hilbert–Kunz 重数

Derivation module and the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve

针对参数对应等差数列的射影单项式曲线坐标环，本文显式构造其导子模的极小生成元，并给出 Hilbert–Kunz 重数公式。

Parametrizing Arf numerical semigroups

We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to six.

Apéry sets and the ideal class monoid of a numerical semigroup

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

Symmetric and Pseudo-Symmetric Numerical Semigroups via Young Diagrams and Their Semigroup Rings

This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric semigroups into an over semigroup and its dual. It is also given exactly for what kind of numerical semigroup $S$, the semigroup ring $\Bbbk[\![S]\!]$ has at least one Gorenstein subring and has at least one Kunz subring.

Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity $m$ in terms of their $m$-Apéry sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a…

通过特殊化研究数值半群代数上的无限自由消解

Infinite free resolutions over numerical semigroup algebras via specialization

本文证明数值半群代数上基域的无限自由消解结构遵循 Kunz 锥面分层，为分类 Golodness 和 Poincaré 级数有理性提供组合方法，并完整分类了 m=4 的情形。

The tree of irreducible numerical semigroups with fixed Frobenius number

In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating 0-1 vectors with as many component as the Frobenius number.

The set of numerical semigroups of a given genus

In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based in the translation of a numerical semigroup to its so-called Kunz-coordinates vector.

Irreducible numerical semigroups with multiplicity three and four

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in (Blanco-Puerto, 2011). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups as those that are generated by…

A semilattice structure for the set of numerical semigroups with fixed Frobenius number

We present a procedure to enumerate the whole set of numerical semigroups with a given Frobenius number F, S(F). The methodology is based on the construction of a partition of S(F) by a congruence relation. We identify exactly one irreducible and one homogeneous numerical semigroup at each class in the relation, and from those two elements we reconstruct the whole class. An alternative more efficient method is proposed based on the use of the Kunz-coordinates vectors of the elements in S(F).

通过Apéry特殊化得到数值半群代数的极小自由分解

Minimal free resolutions of numerical semigroup algebras via Apéry specialization

本文基于Kunz关于数值半群由多面锥整点参数化的结论，构造了toric理想的自由分解，其结构对锥固定面相对内部参数化的半群一致，且在最大嵌入维数时达到极小性。

Integer Programming and m-irreducibility of numerical semigroups

This paper addresses the problem of decomposing a numerical semigroup into m-irreducible numerical semigroups. The problem originally stated in algebraic terms is translated, introducing the so called Kunz-coordinates, to resolve a series of several discrete optimization problems. First, we prove that finding a minimal m-irreducible decomposition is equivalent to solve a multiobjective linear integer problem. Then, we restate that problem as the problem of finding all the optimal solutions of a finite number of single objective integer linear problems plus a set covering problem. Finally, we p…

几乎对称的几乎完全交数值半群的结构

The structure of almost symmetric almost complete intersection numerical semigroups

本文证明几乎对称且几乎完全交的数值半群具有级联矩阵结构，并由此得到刚性定理、级联多项式及Herzog–Watanabe问题的肯定解答。