Arf good semigroups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850182 ◽  
Author(s):  
Giuseppe Zito
Keyword(s):  
Numerical Semigroup ◽  
Arf Numerical Semigroup ◽  
Good Semigroup ◽  
Arf 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.

scholarly journals Parametrizing Arf numerical semigroups

2017 ◽  
Vol 16 (11) ◽  
pp. 1750209 ◽  
Author(s):  
P. A. García-Sánchez ◽  
B. A. Heredia ◽  
H. İ. Karakaş ◽  
J. C. Rosales
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Arf Numerical Semigroup

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 7.

On the numerical semigroup generated by {bn+1+i+bn+i−1b−1∣i∈N}$\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$

2020 ◽  
Vol 30 (4) ◽  
pp. 257-264
Author(s):  
Ze Gu
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Embedding Dimension ◽  
Order Relations ◽  
Positive Integers

AbstractLet b, n be two positive integers such that b ≥ 2, and S(b, n) be the numerical semigroup generated by $\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$. Applying two order relations, we give formulas for computing the embedding dimension, the Frobenius number, the type and the genus of S(b, n).

scholarly journals On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

2018 ◽  
Vol 86 (12) ◽  
pp. 2893-2916 ◽  
Author(s):  
José I. Farrán ◽  
Pedro A. García-Sánchez ◽  
Benjamín A. Heredia
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Geometry Codes ◽  
Rao Distance ◽  
Arf Semigroups

scholarly journals On the Cohen-Macaulay Property for Quadratic Tangent Cones

10.37236/5793 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Dumitru I. Stamate
Keyword(s):  
Tangent Cone ◽  
Gröbner Basis ◽  
Numerical Semigroup ◽  
Tangent Cones ◽  
Grobner Basis

Let $H$ be an $n$-generated numerical semigroup such that its tangent cone $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadratic relations. We show that if $n<5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Cohen-Macaulay, and for $n=5$ we explicitly describe the semigroups $H$ such that $\operatorname{gr}_\mathfrak{m} K[H]$ is not Cohen-Macaulay. As an application we show that if the field $K$ is algebraically closed and of characteristic different from two, and $n\leq 5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gröbner basis.

scholarly journals Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

2015 ◽  
Vol 25 (06) ◽  
pp. 1043-1053 ◽  
Author(s):  
Francesco Strazzanti
Keyword(s):  
Positive Integer ◽  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Minimal Genus

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ ℕ | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is [Formula: see text], where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

scholarly journals Numerical semigroups in a problem about cost-effective transport

2017 ◽  
Vol 29 (2) ◽  
pp. 329-345 ◽  
Author(s):  
Aureliano M. Robles-Pérez ◽  
José Carlos Rosales
Keyword(s):  
Nonnegative Integer ◽  
Numerical Semigroup ◽  
Cost Effective ◽  
Numerical Semigroups ◽  
Integer Coefficients ◽  
Effective Transport ◽  
System Of Inequalities ◽  
Algorithmic Process

AbstractLet ${{\mathbb{N}}}$ be the set of nonnegative integers. A problem about how to transport profitably an organized group of persons leads us to study the set T formed by the integers n such that the system of inequalities, with nonnegative integer coefficients,$a_{1}x_{1}+\cdots+a_{p}x_{p}<n<b_{1}x_{1}+\cdots+b_{p}x_{p}$has at least one solution in ${{\mathbb{N}}^{p}}$. We will see that ${T\cup\{0\}}$ is a numerical semigroup. Moreover, we will show that a numerical semigroup S can be obtained in this way if and only if ${\{a+b-1,a+b+1\}\subseteq S}$, for all ${a,b\in S\setminus\{0\}}$. In addition, we will demonstrate that such numerical semigroups form a Frobenius variety and we will study this variety. Finally, we show an algorithmic process in order to compute T.

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