Frobenius R-Variety of the Numerical Semigroups Contained in a Given One

Abstract

Let $\bigtriangleup$ be a numerical semigroup and ${\mathrm{R}}(\bigtriangleup )=\left\{ S ~|~ S ~\text {is a} \text {numericalsemigroup and} ~S\subseteq \bigtriangleup \right\}$. We prove that ${\mathrm{R}}(\bigtriangleup )$ is Frobenius R-variety that can be arranged in a tree rooted in $\bigtriangleup$. We introduce the concepts of Frobenius and genus number of S restricted to $\bigtriangleup$ (respectively $\mathrm{F}_{\bigtriangleup }(S)$ and ${\mathrm{g}}_{\bigtriangleup }(S)$). We give formulas for $\mathrm{F}_{\bigtriangleup }(S)$, ${\mathrm{g}}_{\bigtriangleup }(S)$ and generalizations of the Amorós’s and Wilf’s conjecture. Moreover, we will show that most of the results of irreducibility can be generalized to ${\mathrm{R}}(\bigtriangleup )$-irreducibility.