Semigroups with fixed multiplicity and embedding dimension

Juan Ignacio Garcı́a-Garcı́a, Daniel Marı́n-Aragón, María Ángeles Moreno-Frías, José Carlos Rosales, Alberto Vigneron-Tenorio


Given m ∈ ℕ, a numerical semigroup with multiplicity m is called a packed numerical semigroup if its minimal generating set is included in {m, m + 1, …, 2m − 1}. In this work, packed numerical semigroups are used to build the set of numerical semigroups with a given multiplicity and embedding dimension, and to create a partition of this set. Wilf’s conjecture is verified in the tree associated to some packed numerical semigroups. Furthermore, given two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. We also compute the semigroups where these minimal values are achieved.


Embedding dimension, Frobenius number, genus, multiplicity, numerical semigroup

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ISSN: 1855-3974

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