scholarly journals Quasi-Ordinarization Transform of a Numerical Semigroup

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1084
Author(s):  
Maria Bras-Amorós ◽  
Hebert Pérez-Rosés ◽  
José Miguel Serradilla-Merinero
Keyword(s):  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Fixed Genus

In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms.

scholarly journals Modular Frobenius pseudo-varieties

2021 ◽  
Author(s):  
Aureliano M. Robles-Pérez ◽  
José Carlos Rosales
Keyword(s):  
Finite Subset ◽  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Fixed Genus

AbstractIf $$m \in {\mathbb {N}} \setminus \{0,1\}$$ m ∈ N \ { 0 , 1 } and A is a finite subset of $$\bigcup _{k \in {\mathbb {N}} \setminus \{0,1\}} \{1,\ldots ,m-1\}^k$$ ⋃ k ∈ N \ { 0 , 1 } { 1 , … , m - 1 } k , then we denote by $$\begin{aligned} {\mathscr {C}}(m,A) =&\{ S\in {\mathscr {S}}_m \mid s_1+\cdots +s_k-m \in S \text { if } (s_1,\ldots ,s_k)\in S^k \text { and } \\ {}&\qquad (s_1 \bmod m, \ldots , s_k \bmod m)\in A \}. \end{aligned}$$ C ( m , A ) = { S ∈ S m ∣ s 1 + ⋯ + s k - m ∈ S if ( s 1 , … , s k ) ∈ S k and ( s 1 mod m , … , s k mod m ) ∈ A } . In this work we prove that $${\mathscr {C}}(m,A)$$ C ( m , A ) is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $${\mathscr {C}}(m,A)$$ C ( m , A ) and to compute all the elements of $${\mathscr {C}}(m,A)$$ C ( m , A ) with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $${\mathscr {C}}(m,A)$$ C ( m , A ) when $$A=\{1,\ldots ,m-1\}^3$$ A = { 1 , … , m - 1 } 3 , $$A=\{(1,1),\ldots ,(m-1,m-1)\}$$ A = { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , and $$A=\{1,\ldots ,m-1\}^2 \setminus \{(1,1),\ldots ,(m-1,m-1)\}$$ A = { 1 , … , m - 1 } 2 \ { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , respectively.

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.

scholarly journals The Numerical Semigroup of Phrases' Lengths in a Simple Alphabet

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Aureliano M. Robles-Pérez ◽  
José Carlos Rosales
Keyword(s):  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Starting Point ◽  
The Family

Let𝒜be an alphabet with two elements. Considering a particular class of words (the phrases) over such an alphabet, we connect with the theory of numerical semigroups. We study the properties of the family of numerical semigroups which arise from this starting point.

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.

scholarly journals Measuring primality in numerical semigroups with embedding dimension three

2015 ◽  
Vol 15 (01) ◽  
pp. 1650007 ◽  
Author(s):  
S. T. Chapman ◽  
P. A. García-Sánchez ◽  
Z. Tripp ◽  
C. Viola
Keyword(s):  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Catenary Degree

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.

scholarly journals Characterization of Graphs Associated with Numerical Semigroups

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 557 ◽  
Author(s):  
Muhammad Ahsan Binyamin ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Nida Munawar Khan ◽  
Adnan Aslam ◽  
Yongsheng Rao
Keyword(s):  
Undirected Graph ◽  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Vertex Set

Let Γ be a numerical semigroup. We associate an undirected graph G ( Γ ) with a numerical semigroup Γ with vertex set { v i : i ∈ N \ Γ } and edge set { v i v j ⇔ i + j ∈ Γ } . In this article, we discuss the connectedness, diameter, girth, and some other related properties of the graph G ( Γ ) .

scholarly journals Embedding the set of nondivisorial ideals of a numerical semigroup into ℕn

2018 ◽  
Vol 17 (11) ◽  
pp. 1850205 ◽  
Author(s):  
Dario Spirito
Keyword(s):  
Partial Order ◽  
Numerical Semigroup ◽  
Natural Partial Order ◽  
Numerical Semigroups ◽  
Star Operations

The set [Formula: see text] of the classes of nondivisorial ideals of a numerical semigroup [Formula: see text] can be endowed with a natural partial order induced by the set of star operations on [Formula: see text]. We study embeddings of [Formula: see text] into [Formula: see text], specializing on three families of numerical semigroups with radically different behavior.

Delta sets for nonsymmetric numerical semigroups with embedding dimension three

2018 ◽  
Vol 30 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Pedro A. García-Sánchez ◽  
David Llena ◽  
Alessio Moscariello
Keyword(s):  
Fast Algorithm ◽  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Delta Set

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.

scholarly journals Union of Sets of Lengths of Numerical Semigroups

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1789
Author(s):  
J. I. García-García ◽  
D. Marín-Aragón ◽  
A. Vigneron-Tenorio
Keyword(s):  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Almost Periodic ◽  
Sets Of Lengths

Let S=⟨a1,…,ap⟩ be a numerical semigroup, let s∈S and let Z(s) be its set of factorizations. The set of lengths is denoted by L(s)={L(x1,⋯,xp)∣(x1,⋯,xp)∈Z(s)}, where L(x1,⋯,xp)=x1+⋯+xp. The following sets can then be defined: W(n)={s∈S∣∃x∈Z(s)suchthatL(x)=n}, ν(n)=⋃s∈W(n)L(s)={l1<l2<⋯<lr} and Δν(n)={l2−l1,…,lr−lr−1}. In this paper, we prove that the function Δν:N→P(N) is almost periodic with period lcm(a1,ap).

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