Classes of numerical semigroups with embedding dimension 3: an algorithm for computing the Frobenius number

2021 ◽  
Vol 15 (4) ◽  
pp. 171-179
Author(s):  
Violeta Angjelkoska ◽  
Donco Dimovski ◽  
Irena Stojmenovska
Keyword(s):  
Frobenius Number ◽  
Numerical Semigroups ◽  
Embedding Dimension

An approach in computing the Frobenius number of numerical semigroups with embedding dimension equal to 3

2021 ◽  
Vol 15 (4) ◽  
pp. 181-190
Author(s):  
Violeta Angjelkoska ◽  
Donco Dimovski ◽  
Irena Stojmenovska
Keyword(s):  
Frobenius Number ◽  
Numerical Semigroups ◽  
Embedding Dimension

scholarly journals Random Numerical Semigroups and a Simplicial Complex of Irreducible Semigroups

10.37236/7796 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Jesus De Loera ◽  
Christopher O'Neill ◽  
Dane Wilburne
Keyword(s):  
Simplicial Complex ◽  
Random Graphs ◽  
Probabilistic Model ◽  
Frobenius Number ◽  
Threshold Function ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Shellable Simplicial Complex

We examine properties of random numerical semigroups  under a probabilistic model inspired by the Erdos-Renyi model for random graphs. We provide a threshold function for cofiniteness, and bound the expected embedding dimension, genus, and Frobenius number of random semigroups.  Our results follow, surprisingly, from the construction of a very natural shellable simplicial complex whose facets are in bijection with irreducible numerical semigroups of a fixed Frobenius number and whose $h$-vector determines the probability that a particular element lies in the semigroup.

The Frobenius problem for a class of numerical semigroups

2017 ◽  
Vol 13 (05) ◽  
pp. 1335-1347 ◽  
Author(s):  
Ze Gu ◽  
Xilin Tang
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Frobenius Problem ◽  
Positive Integers

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].

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).

Numerical semigroups whose fractions are of maximal embedding dimension

2010 ◽  
Vol 82 (3) ◽  
pp. 412-422 ◽  
Author(s):  
David E. Dobbs ◽  
Harold J. Smith
Keyword(s):  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Maximal Embedding Dimension

scholarly journals On the Frobenius number of certain numerical semigroups

2021 ◽  
Author(s):  
M. Hellus ◽  
A. Rechenauer ◽  
R. Waldi
Keyword(s):  
Frobenius Number ◽  
Numerical Semigroups

COUNTING NUMERICAL SEMIGROUPS WITH SHORT GENERATING FUNCTIONS

2011 ◽  
Vol 21 (07) ◽  
pp. 1217-1235 ◽  
Author(s):  
VÍCTOR BLANCO ◽  
PEDRO A. GARCÍA-SÁNCHEZ ◽  
JUSTO PUERTO
Keyword(s):  
Generating Function ◽  
Polynomial Time ◽  
Generating Functions ◽  
Time Complexity ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Polynomial Time Complexity ◽  
Theoretical Results

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

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.

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