Article

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.

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