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On Free Numerical Semigroups and the Construction of Minimal Telescopic Sequences [article]

Caleb M. Shor
2019 arXiv   pre-print
In the process, we will examine some operations on and constructions of telescopic sequences in general.  ...  In particular, given a free numerical semigroup we can construct a telescopic generating sequence which is minimal.  ...  Acknowledgments The author wishes to thank the anonymous referee for substantially helping to improve the presentation of this paper in many ways.  ... 
arXiv:1806.11381v2 fatcat:hsr7qw6yxvedbk5n5s22iayl2i

Constructing the set of complete intersection numerical semigroups with a given Frobenius number [article]

Abdallah Assi, Pedro A. García-Sánchez
2013 arXiv   pre-print
We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to  ...  The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.  ...  in the set of free numerical semigroups), and that of numerical semigroups associated to an irreducible plane curve singularity (these are a particular case of telescopic numerical semigroups).  ... 
arXiv:1204.4258v3 fatcat:lcsafvt7bjbi5enmlridcr5osu

Reflective numerical semigroups [article]

Caleb M. Shor
2022 arXiv   pre-print
With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius  ...  Equivalently, a reflective numerical semigroup has one gap in each residue class modulo g. In this paper, we give an explicit description for all reflective numerical semigroups.  ...  A numerical semigroup is free if it is generated by a telescopic sequence. However, the corresponding telescopic set need not be minimal.  ... 
arXiv:2207.00051v1 fatcat:rad2nqksb5aizmspbencrf7e54

Generalized Strongly Increasing Semigroups

E. R. García Barroso, J. I. García-García, A. Vigneron-Tenorio
2021 Mathematics  
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.  ...  Acknowledgments: The authors thank the referees for their helpful observations. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/math9121370 fatcat:ltjap5hqjrdidkftxv3opeepdu

The second Feng–Rao number for codes coming from telescopic semigroups

José I. Farrán, Pedro A. García-Sánchez, Benjamín A. Heredia, Micah J. Leamer
2017 Designs, Codes and Cryptography  
In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup.  ...  To achieve this result we first study the behavior of Apéry sets under gluings of numerical semigroups.  ...  Additionally, recall that this bound equals the one given by the actual second Feng-Rao distance in most cases and specifically for all a ≥ c.  ... 
doi:10.1007/s10623-017-0426-5 fatcat:ow7k4hxpgrdzlpjbdzp5xfkmd4

The second Feng-Rao number for codes coming from telescopic semigroups [article]

José I. Farrán and P. A. García-Sánchez and B. A. Heredia and M. J. Leamer
2017 arXiv   pre-print
In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup.  ...  To achieve this result we first study the behavior of Ap\'ery sets under gluings of numerical semigroups.  ...  The fourth author would like to thank Marco D'Anna and the rest of orginizers of the INdAM meeting: International meeting on numerical semigroups -Cortona 2014.  ... 
arXiv:1603.09301v2 fatcat:3r223xnxvnalzdvysb2ntcaske

Generalized strongly increasing semigroups [article]

E.R. García Barroso, J.I. García-García, A. Vigneron-Tenorio
2020 arXiv   pre-print
In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families of semigroups and we give explicitly 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.  ...  The new Frobenius numbers are 7980 and 26460. Some GSI-semigroups with these Frobenius numbers are: S 12 ⊕ 13,652 N and S 12 ⊕ 17,486 N for 7980, and S 12 ⊕ 13,2192 N and S 12 ⊕ 17,1641 N for 26460.  ... 
arXiv:2003.13381v1 fatcat:ggbvl34v4jggnlnw3o3tu724pe

numericalsgps, a GAP package for numerical semigroups

M. Delgado, P. A. García-Sánchez
2016 ACM Communications in Computer Algebra  
This manuscript is a survey of what the package does, and at the same time intends to gather the trending topics on numerical semigroups.  ...  The package numericalsgps performs computations with and for numerical and affine semigroups.  ...  sets (α, β and γ-rectangular, see [23] ); and the type sequence of a numerical semigroup [7] .  ... 
doi:10.1145/3003653.3003656 fatcat:fwirsvjh5rey5ctwqxtojjnawa

numericalsgps, a GAP package for numerical semigroups

M. Delgado, P. A. García-Sánchez
2016 ACM Communications in Computer Algebra  
This manuscript is a survey of what the package does, and at the same time of the trending topics on numerical semigroups.  ...  The package numericalsgps performs computations with and for numerical semigroups. Recently also affine semigroups are admitted as objects for calculations.  ...  sets (α, β and γ-rectangular, see [23] ); and the type sequence of a numerical semigroup [7] .  ... 
doi:10.1145/2930964.2930966 fatcat:blcnrbgmkfajbmhxn5qo2ndrv4

δ-sequences and Evaluation Codes defined by Plane Valuations at Infinity [article]

C. Galindo, F.Monserrat
2008 arXiv   pre-print
We also give algorithms to construct an unlimited number of δ-sequences of the different existing types, and so this paper provides the tools to know and use a new large set of codes.  ...  We introduce the concept of δ-sequence. A δ-sequence Δ generates a well-ordered semigroup S in Z^2 or R.  ...  Now the semigroup spanned by {α i } r−1 i=1 behaves as the one generated by the elements, except the last one, of a telescopic semigroup.  ... 
arXiv:0704.3536v2 fatcat:dlzyexkl3bgnnbdpwbj5ktrrei

δ-sequences and evaluation codes defined by plane valuations at infinity

Carlos Galindo, Francisco Monserrat
2008 Proceedings of the London Mathematical Society  
It is also worth adding that the obtained image semigroups have a behavior close to the one of telescopic semigroups and that the so-called approximates of the valuation at infinity (see Definition 4.6  ...  We introduce the concept of δ-sequence. A δ-sequence ∆ generates a well-ordered semigroup S in Z 2 or R.  ...  Now the semigroup spanned by {α i } r−1 i=1 behaves as the one generated by the elements, except the last one, of a telescopic semigroup.  ... 
doi:10.1112/plms/pdn042 fatcat:ixmx5oq2lzak7mjd3cunj2ayei

Uniquely presented finitely generated commutative monoids

Pedro García-Sánchez, Ignacio Ojeda
2010 Pacific Journal of Mathematics  
We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented.  ...  A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation.  ...  Part of this work was done during a visit of García-Sánchez to the University of Extremadura financed by the Plan Propio 2009 of the University of Extremadura.  ... 
doi:10.2140/pjm.2010.248.91 fatcat:m2scunvfvnfz7lsh4qgb433udq

Huneke-Wiegand Conjecture for Complete Intersection Numerical Semigroup Rings [article]

Pedro A. Garcia-Sanchez, Micah J. Leamer
2012 arXiv   pre-print
We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup  ...  Free numerical semigroups are Huneke-Wiegand. Free numerical semigroups include telescopic numerical semigroups and numerical semigroups associated to an irreducible planar curve singularity.  ...  Let Γ = a 1 Γ 1 + a 2 Γ 2 be a gluing of two numerical semigroups. Next we will focus on sequences with two steps having the middle term and at least one other term in Ap(Γ, a 1 a 2 ).  ... 
arXiv:1211.4554v1 fatcat:ppuhjjyh4nfxfkas64lil6fr5i

Numerical semigroups and applications [article]

Abdallah Assi, Pedro A. García-Sánchez
2014 arXiv   pre-print
The aim of this manuscript is to give some basic notions related to numerical semigroups, and from these on the one hand describe a classical application to the study of singularities of plane algebraic  ...  curves, and on the other, show how numerical semigroups can be used to obtain handy examples of nonunique factorization invariants.  ...  The are many other nonunique factorization invariants that can be defined on any numerical semigroup.  ... 
arXiv:1411.6093v1 fatcat:6jtcitcqp5hgpe4m3fouuanotm

A Graphical Approach to Finding the Frobenius Number, Genus and Hilbert Series of a Numerical Semigroup [article]

Alexandru Pascadi
2018 arXiv   pre-print
This approach applies to many of the cases considered in literature, including semigroups generated by arithmetic and geometric sequences, compound sequences, progressions of the form a^n, a^n + a, ...  ...  This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem.  ...  The author is deeply grateful to Professor Terence Tao for his helpful insights and suggestions during the development of this article.  ... 
arXiv:1712.02522v3 fatcat:hvbvcfwiinfk5lhth3eamkloqe
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