Numerical semigroup tree of multiplicities 4 and 5

Abstract

A numerical semigroup $S$ is a cofinite submonoid of the nonnegative integers under addition. The cardinality of the complement of $S$ in the nonnegative integers is called the genus. The smallest nonzero element of $S$ is the multiplicity of $S$. There is an extensive literature about the tree of numerical semigroups, which has been used to count numerical semigroups by genus, yet the structure of the tree itself has not been described in the literature. In this paper, we completely describe the structure of the subtrees of the numerical semigroup tree of multiplicities 4 and 5. We conclude with an application of these numerical semigroup trees’ structure.