Numerical semigroups with a monotonic Apéry set. (English) Zbl 1081.20071

Summary: We study numerical semigroups \(S\) with the property that if \(m\) is the multiplicity of \(S\) and \(w(i)\) is the least element of \(S\) congruent with \(i\) modulo \(m\), then \(0<w(1)<\cdots<w(m-1)\). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups.


20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems
11D75 Diophantine inequalities
Full Text: DOI EuDML


[1] F. Ajili and E. Contejean: Avoiding slack variables in the solving of linear Diophantine equations and inequations. Principles and practice of constraint programming. Theoret. Comput. Sci. 173 (1997), 183–208. · Zbl 0903.11033
[2] R. Apery: Sur les branches superlineaires des courbes algebriques. C. R. Acad. Sci. Paris 222 (1946). · Zbl 0061.33604
[3] V. Barucci, D. E. Dobbs and M. Fontana: Maximality Properties in Numerical Semi-groups and Applications to One-Dimensional Analytically Irreducible Local Domains. Memoirs of the Amer. Math. Soc. Vol. 598. 1997. · Zbl 0868.13003
[4] J. Bertin and P. Carbonne: Semi-groupes d’entiers et application aux branches. J. Algebra 49 (1987), 81–95. · Zbl 0498.14016
[5] A. Brauer: On a problem of partitions. Amer. J. Math. 64 (1942), 299–312. · Zbl 0061.06801
[6] H. Bresinsky: On prime ideals with generic zero x i = t ni . Proc. Amer. Math. Soc. 47 (1975), 329–332. · Zbl 0296.13007
[7] E. Contejean and H. Devie: An efficient incremental algorithm for solving systems of linear diophantine equations. Inform. and Comput. 113 (1994), 143–172. · Zbl 0809.11015
[8] C. Delorme: Sous-monoides d’intersection complete de \(\mathbb{N}\). Ann. Scient. Ecole Norm. Sup. 9 (1976), 145–154. · Zbl 0325.20065
[9] R. Froberg, C. Gottlieb and R. Haggkvist: Semigroups, semigroup rings and analytically irreducible rings. Reports Dpt. of Mathematics. University of Stockholm, Vol. 1, 1986.
[10] R. Froberg, C. Gottlieb and R. Haggkvist: On numerical semigroups. Semigroup Forum 35 (1987), 63–83. · Zbl 0614.10046
[11] P. A. Garcia-Sanchez and J. C. Rosales: Numerical semigroups generated by intervals. Pacific J. Math. 191 (1999), 75–83. · Zbl 1009.20069
[12] R. Gilmer: Commutative Semigroup Rings. The University of Chicago Press, 1984.
[13] J. Herzog: Generators and relations of abelian semigroups and semigroup rings. Manuscripta Math 3 (1970), 175–193. · Zbl 0211.33801
[14] E. Kunz: The value-semigroup of a one-dimensional Gorenstein ring. Proc. Amer. Math. Soc. 25 (1973), 748–751. · Zbl 0197.31401
[15] J. L. Ramirez Alfonsin: The Diophantine Frobenius problem. Forschungsintitut fur Diskrete Mathematik, Bonn, Report No.00893. 2000.
[16] J. L. Ramirez Alfonsin: The Diophantine Frobenius problem, manuscript.
[17] J. C. Rosales: On numerical semigroups. Semigroup Forum 52 (1996), 307–318. · Zbl 0853.20041
[18] J. C. Rosales: On symmetric numerical semigroups. J. Algebra 182 (1996), 422–434. · Zbl 0856.20043
[19] J. C. Rosales and M. B. Branco: Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups. J. Pure Appl. Algebra 171 (2002), 303–314. · Zbl 1006.20043
[20] J. C. Rosales and P. A. Garcia-Sanchez: Finitely Generated Commutative Monoids. Nova Science Publishers, New York, 1999.
[21] J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and M. B. Branco: Systems of inequalities and numerical semigroups. J. London Math. Soc. 65 (2002), 611–623. · Zbl 1022.20032
[22] J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and J. M. Urbano-Blanco: Proportionally modular Diophantine inequalities. J. Number Theory 103 (2003), 281–294. · Zbl 1039.20036
[23] E. S. Selmer: On a linear Diophantine problem of Frobenius. J. Reine Angew. Math. 293/294 (1977), 1–17. · Zbl 0349.10009
[24] K. Watanabe: Some examples of one dimensional Gorenstein domains. Nagoya Math. J. 49 (1973), 101–109. · Zbl 0257.13024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.