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The Type of a Good Semigroup and the Almost Symmetric Condition

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Abstract

We study the type and the almost symmetric condition for good subsemigoups of \({\mathbb {N}}^2\), a class of semigroups containing the value semigroups of curve singularities with two branches. We define the type in term of a partition of a specific set associated to the semigroup and we show that this definition generalizes the well-known notion of type of a numerical semigroup and has a good behaviour with respect to the corresponding concept for algebroid curves. Then, we study almost symmetric good semigroups, their connections with maximal embedding dimension good semigroups and their Apéry set, generalizing to this context several existent known results.

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References

  1. Barucci, V., D’Anna, M., Fröberg, R.: Analytically unramified one-dimensional semilocal rings and their value semigroups. J. Pure Appl. Algebra 147, 215–254 (2000)

    Article  MathSciNet  Google Scholar 

  2. Barucci, V., D’Anna, M., Fröberg, R.: The Apéry algorithm for a plane singularity with two branches. Beitrage zur Algebra und Geometrie 46(1), 1–18 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Barucci, V., D’Anna, M., Fröberg, R.: The semigroup of values of a one-dimensional local ring with two minimal primes. Commun. Algebra 28(8), 607–3633 (2000)

    Article  MathSciNet  Google Scholar 

  4. Barucci, V., Fröberg, R.: One-dimensional almost Gorenstein rings. J. Algebra 188, 418–442 (1997)

    Article  MathSciNet  Google Scholar 

  5. Campillo, A., Delgado, F., Gusein-Zade, S.M.: On generators of the semigroup of a plane curve singularity. J. Lond. Math. Soc. (2) 60, 420–430 (1999)

    Article  MathSciNet  Google Scholar 

  6. Campillo, A., Delgado, F., Kiyek, K.: Gorenstein properties and symmetry for one-dimensional local Cohen–Macaulay rings. Manuscr. Math. 83, 405–423 (1994)

    Article  Google Scholar 

  7. Carvalho, E., Hernandes, M.E.: The semiring of values of an algebroid curve (2017). arXiv:1704.04948v1

  8. D’Anna, M.: The canonical module of a one-dimensional reduced local ring. Commun. Algebra 25, 2939–2965 (1997)

    Article  MathSciNet  Google Scholar 

  9. D’Anna, M., Garcia Sanchez, P., Micale, V., Tozzo, L.: Good semigroups of \(N^n\). Int. J. Algebra Comput. 28, 179–206 (2018)

    Article  Google Scholar 

  10. D’Anna, M., Guerrieri, L., Micale, V.: The Apéry Set of a Good Semigroup (2018). arXiv:1812.02064

  11. Delgado, F.: The semigroup of values of a curve singularity with several branches. Manuscr. Math. 59, 347–374 (1987)

    Article  MathSciNet  Google Scholar 

  12. Delgado, F.: Gorenstein curves and symmetry of the semigroup of value. Manuscr. Math. 61, 285–296 (1988)

    Article  MathSciNet  Google Scholar 

  13. García, A.: Semigroups associated to singular points of plane curves. J. Reine Angew. Math. 336, 165–184 (1982)

    MathSciNet  MATH  Google Scholar 

  14. Goto, S., Matsuoka, N., Phuong, T.T.: Almost Gorenstein rings. J. Algebra 379, 355–381 (2013)

    Article  MathSciNet  Google Scholar 

  15. Goto, S., Takahashi, R., Taniguchi, N.: Almost Gorenstein rings—towards a theory of higher dimension. J. Pure Appl. Algebra 219(7), 2666–2712 (2015)

    Article  MathSciNet  Google Scholar 

  16. Herzog, J., Kunz, E.: Die Wertehalbgruppe eines lokalen Rings der Dimension 1, Sitz. Ber. Heidelberger Akad. Wiss, pp. 27–43 (1971)

  17. Jäger, J.: Längenberechnung und kanonische Ideale in eindimensionalen Ringen. Arch. Math. (Basel) 29(5), 504–512 (1977)

    Article  MathSciNet  Google Scholar 

  18. Korell, P., Schulze, M., Tozzo, L.: Duality on value semigroups. J. Commut. Algebra 11(1), 81–129 (2019)

    Article  MathSciNet  Google Scholar 

  19. Kunz, E.: The value-semigroup of a one-dimensional Gorenstein ring. Proc. Am. Math. Soc. 25, 748–751 (1970)

    Article  MathSciNet  Google Scholar 

  20. Maugeri, N., Zito, G.: Embedding dimension of a good semigroup (2019). arXiv:1903.02057

  21. Nari, H.: Symmetries on almost symmetric numerical semigroups. Semigroup Forum 86, 140–154 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria, 6, 95125, Catania, Italy

    Marco D’Anna, Lorenzo Guerrieri & Vincenzo Micale

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  1. Marco D’Anna
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  2. Lorenzo Guerrieri
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  3. Vincenzo Micale
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Correspondence to Marco D’Anna.

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D’Anna, M., Guerrieri, L. & Micale, V. The Type of a Good Semigroup and the Almost Symmetric Condition. Mediterr. J. Math. 17, 28 (2020). https://doi.org/10.1007/s00009-019-1467-y

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  • DOI: https://doi.org/10.1007/s00009-019-1467-y

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