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On Free Numerical Semigroups and the Construction of Minimal Telescopic Sequences
[article]
2019
arXiv
pre-print
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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]
2013
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
2021
Mathematics
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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
2017
Designs, Codes and Cryptography
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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]
2017
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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
2016
ACM Communications in Computer Algebra
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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
2016
ACM Communications in Computer Algebra
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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]
2008
arXiv
pre-print
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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
2008
Proceedings of the London Mathematical Society
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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
2010
Pacific Journal of Mathematics
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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]
2012
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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]
2018
arXiv
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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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