The tree of irreducible numerical semigroups with fixed Frobenius number

Irreducible Numerical Semigroups

We give a characterization for irreducible numerical semi-groups. From this characterization we obtain that every irre-ducible numerical semigroup is either a symmetric or pseudo-symmetric numerical semigroup. We study the minimal presentations of an irreducible numerical semigroup. Separately, we deal with the cases of maximal embedding dimension and multiplicity 3 and 4.

The Frobenius problem for numerical semigroups

In this paper, we characterize those numerical semigroups containing 〈n1, n2〉. From this characterization, we give formulas for the genus and the Frobenius number of a numerical semigroup. These results can be used to give a method for computing the genus and the Frobenius number of a numerical semigroup with embedding dimension three in terms of its minimal system of generators.

Pacific Journal of Mathematics IRREDUCIBLE NUMERICAL SEMIGROUPS

Irreducible Numerical Semigroups Having Toms Decomposition

In this paper we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.

Exploring the tree of numerical semigroups

In this paper we describe an algorithm visiting all the numerical semigroups up to a given genus using a new representation of numerical semigroups.