The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number Moreno Frías, María Ángeles Rosales González, José Carlos Numerical semigroup Ratio-covariety Frobenius number Genus Multiplicity Algorithm 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. 2025-07-29T11:36:45Z 2025-07-29T11:36:45Z 2024-09-18 journal article Moreno Frias, M. A., & Rosales, J. C. (2025). The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number. International Electronic Journal of Algebra, 38(38), 12-28. https://doi.org/10.24330/ieja.1575996 https://hdl.handle.net/10481/105825 10.24330/ieja.1575996 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional International Electronic Journal of Algebra