Irreducible numerical semigroups

Abstract

Symmetric numerical semigroups are probably the numerical semigroups that have been most studied in the literature. The motivation and introduction of these semigroups is due mainly to Kunz, who in his manuscript [44] proves that a onedimensional analytically irreducible Noetherian local ring is Gorenstein if and only if its value semigroup is symmetric. Symmetric numerical semigroups always have odd Frobenius number. The translation of this concept for numerical semigroups with even Frobenius number motivates the definition of pseudo-symmetric numerical semigroups. In [5] it is shown that these semigroups also have their interpretation in one-dimensional local rings, since a numerical semigroup is pseudo-symmetric if and only if its semigroup ring is a Kunz ring.