scholarly journals Characterizations of numerical semigroup complements via Apéry sets

2018 ◽  
Vol 98 (1) ◽  
pp. 31-47
Author(s):  
T. Alden Gassert ◽  
Caleb McKinley Shor
Keyword(s):  
Numerical Semigroup ◽  
Apéry Sets

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 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.

Compositions of a numerical semigroup

2019 ◽  
Vol 29 (5) ◽  
pp. 345-350
Author(s):  
Ze Gu
Keyword(s):  
Nonnegative Integer ◽  
Numerical Semigroup ◽  
Apéry Set ◽  
Wilf’S Conjecture

Abstract Given a numerical semigroup S, a nonnegative integer a and m ∈ S ∖ {0}, we introduce the set C(S, a, m) = {s + aw(s mod m) | s ∈ S}, where {w(0), w(1), ⋯, w(m – 1)} is the Apéry set of m in S. In this paper we characterize the pairs (a, m) such that C(S, a, m) is a numerical semigroup. We study the principal invariants of C(S, a, m) which are given explicitly in terms of invariants of S. We also characterize the compositions C(S, a, m) that are symmetric, pseudo-symmetric and almost symmetric. Finally, a result about compliance to Wilf’s conjecture of C(S, a, m) is given.

Sign in / Sign up

Export Citation Format

Share Document