Internal numerical semigroups

Abstract

In this paper the tree structure of numerical semigroups is studied. An internal numerical semigroup is a semigroup located in an internal node of the tree. Analogous a leaf numerical semigroup is placed in a leaf node. Internal semigroups with fixed multiplicity, Frobenius number or genus are studied by providing algorithms to construct all of them. Several conjectures are established, for example, in each case (fixed multiplicity, fixed Frobenius number and fixed genus respectively), the results suggest that there are always more internal than leaf numerical semigroups. Finally, numerical semigroups with fixed multiplicity and Frobenius number simultaneously are investigated. In this case with two invariants fixed, moreover closed formulas to count the number of internal and leaf semigroups are provided for some values of multiplicity and Frobenius number.