The Frobenius problem for a class of numerical semigroups

2017 ◽  
Vol 13 (05) ◽  
pp. 1335-1347 ◽  
Author(s):  
Ze Gu ◽  
Xilin Tang

Let [Formula: see text] be two positive integers such that [Formula: see text] and [Formula: see text] the numerical semigroup generated by [Formula: see text]. Then [Formula: see text] is the Thabit numerical semigroup introduced by J. C. Rosales, M. B. Branco and D. Torrão. In this paper, we give formulas for computing the Frobenius number, the genus and the embedding dimension of [Formula: see text].

2020 ◽  
Vol 30 (4) ◽  
pp. 257-264
Author(s):  
Ze Gu

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).


2016 ◽  
Vol 146 (5) ◽  
pp. 1081-1090
Author(s):  
Aureliano M. Robles-Pérez ◽  
José Carlos Rosales

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.


2015 ◽  
Vol 25 (06) ◽  
pp. 1043-1053 ◽  
Author(s):  
Francesco Strazzanti

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.


2017 ◽  
Vol 16 (11) ◽  
pp. 1750209 ◽  
Author(s):  
P. A. García-Sánchez ◽  
B. A. Heredia ◽  
H. İ. Karakaş ◽  
J. C. Rosales

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.


2015 ◽  
Vol 15 (01) ◽  
pp. 1650007 ◽  
Author(s):  
S. T. Chapman ◽  
P. A. García-Sánchez ◽  
Z. Tripp ◽  
C. Viola

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree.


2018 ◽  
Vol 30 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Pedro A. García-Sánchez ◽  
David Llena ◽  
Alessio Moscariello

Abstract We present a fast algorithm to compute the Delta set of a nonsymmetric numerical semigroup with embedding dimension three. We also characterize the sets of integers that are the Delta set of a numerical semigroup of this kind.


Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1370
Author(s):  
E. R. García Barroso ◽  
J. I. García-García ◽  
A. Vigneron-Tenorio

In this work, we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and other families of semigroups and we explicitly give their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a given Frobenius number is provided and the realization of positive integers as Frobenius numbers of GSI-semigroups is studied.


10.37236/410 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
F. Aguiló-Gost ◽  
P. A. García-Sánchez

Let us consider a $3$-numerical semigroup $S=\langle{a,b,N}\rangle$. Given $m\in S$, the triple $(x,y,z)\in\mathbb{N}^3$ is a factorization of $m$ in $S$ if $xa+yb+zN=m$. This work is focused on finding the full set of factorizations of any $m\in S$ and as an application we compute the catenary degree of $S$. To this end, we relate a 2D tessellation to $S$ and we use it as a main tool.


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