The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number

Abstract

In this paper we will show that MED(F, m) = {S | S is a numerical semigroup with maximal embedding dimension, Frobenius number F and multiplicity m} is a ratio-covariety. As a consequence, we present two algorithms: one that computes MED(F, m) and another one that calculates the elements of MED(F, m) with a given genus. If X ⊆ S\(⟨m⟩∪{F +1, →}) for some S ∈ MED(F, m), then there exists the smallest element of MED(F, m) containing X. This element will be denoted by MED(F, m)[X] and we will say that X one of its MED(F, m)-system of generators. We will prove that every element S of MED(F, m) has a unique minimal MED(F, m)-system of generators and it will be denoted by MED(F, m)msg(S). The cardinality of MED(F, m)msg(S), will be called MED(F, m)-rank of S. We will also see in this work, how all the elements of MED(F, m) with a fixed MED(F, m)-rank are.