Abstract
Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈ℕ, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.
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References
Barucci, V., Dobbs, D.E., Fontana, M.: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Mem. Am. Math. Soc. 125(598) (1997)
Bras-Amorós, M., García-Sánchez, P.A.: Patterns on numerical semigroups. Linear Algebra Appl. 414, 652–669 (2006)
Fröberg, R., Gottlieb, C., Häggkvist, R.: On numerical semigroups. Semigroup Forum 35, 63–83 (1987)
Ramírez Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford Lecture Series Math. and Its Appl., vol. 30. Oxford Univ. Press, Oxford (2005)
Robles-Pérez, A.M., Rosales, J.C., Vasco, P.: The doubles of a numerical semigroup. J. Pure Appl. Algebra 213, 387–396 (2009)
Rosales, J.C.: One half of a pseudo-symmetric numerical semigroup. Bull. Lond. Math. Soc. 40, 347–352 (2008)
Rosales, J.C.: Families of numerical semigroups closed under finite intersections and for the Frobenius number. Houst. J. Math. 34, 339–348 (2008)
Rosales, J.C., García-Sánchez, P.A.: Every numerical semigroup is one half of a symmetric numerical semigroup. Proc. Am. Math. Soc. 136, 475–477 (2008)
Rosales, J.C., García-Sánchez, P.A.: Every numerical semigroup is one half of infinitely many symmetric numerical semigroups. Commun. Algebra 36, 2910–2916 (2008)
Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. Developments in Math., vol. 20. Springer, New York (2009)
Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Branco, M.B.: Numerical semigroups with maximal embedding dimension. Int. J. Commut. Rings 2, 47–53 (2003)
Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Urbano-Blanco, J.M.: Proportionally modular diuophantine inequalities. J. Number Theory 103, 281–294 (2003)
Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Branco, M.B.: Arf numerical semigroups. J. Algebra 276, 3–12 (2004)
Smith, H.J.: Numerical semigroups that are fractions of numerical semigroups of maximal embedding dimension. JP J. Algebra Number Theory Appl. 17, 69–96 (2010)
Swanson, I.: Every numerical semigroup is one over d of infinitely many symmetric numerical semigroups. In: Commutative Algebra and its Applications, pp. 383–386. de Gruyter, Berlin (2009)
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Communicated by Jorge Almeida.
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Dobbs, D.E., Smith, H.J. Numerical semigroups whose fractions are of maximal embedding dimension. Semigroup Forum 82, 412–422 (2011). https://doi.org/10.1007/s00233-010-9275-5
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DOI: https://doi.org/10.1007/s00233-010-9275-5