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

Abstract

Let be a numerical semigroup and R( ) = {S | S is a numericalsemigroup and S ⊆ }. We prove that R( ) is Frobenius R- variety that can be arranged in a tree rooted in . We introduce the concepts of Frobenius and genus number of S restricted to (respec-tively F (S) and g (S)). We give formulas for F (S), g (S) and generalizations of the Amor ́os’s and Wilf’s conjecture. Moreover, we will show that most of the results of irreducibility can be generalized to R( )-irreducibility.