Genus of numerical semigroups generated by three elements

Journal of Algebra (Impact Factor: 0.6). 04/2011; 358. DOI: 10.1016/j.jalgebra.2012.02.010
Source: arXiv

ABSTRACT Let H = < a, b, c > be a numerical semigroup generated by three elements and let R = k vertical bar H vertical bar be its semigroup ring over a field k. We assume that H is not symmetric and assume that the defining ideal of R is defined by maximal minors of the matrix ((X alpha)(Y beta)(Z gamma)(Y beta')(Z gamma')(X alpha')). Then we will show that the genus of H is determined by the Frobenius number F(H) and alpha beta gamma or alpha'beta'gamma' In particular, we show that H is pseudo-symmetric if and only if alpha beta gamma = 1 or alpha'beta'gamma' = 1. Also, we will give a simple algorithm to get all the pseudo-symmetric numerical semigroups H = < a. b, c > with given Frobenius number.

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The notion of an almost symmetric numerical semigroup was given by V. Barucci and R. Froberg in J. Algebra, 188, 418-442 (1997). We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H (au) (the dual of M) to be an almost symmetric numerical semigroup. Using these results we give a formula for the multiplicity of an opened modular numerical semigroup. Finally, we show that if H (1) or H (2) is not symmetric, then the gluing of H (1) and H (2) is not almost symmetric.
    Semigroup Forum 11/2011; 86(1). DOI:10.1007/s00233-012-9397-z · 0.38 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in full generality, where $\m$ denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case.


Available from