Papers › The complexity of a numerical semigroup
The complexity of a numerical semigroup
J. I. García-García, M. A. Moreno-Frías, J. C. Rosales, A. Vigneron-Tenorio
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Let S and Δ be numerical semigroups. A numerical semigroup S is an 𝐈(Δ)-{\it semigroup} if S\{0} is an ideal of Δ. We will denote by 𝒥(Δ)={S |S is an\mathbf{I}(\Delta)-semigroup }. We will say that Δ is {\it an ideal extension of } S if S∈𝒥(Δ). In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by 𝒥⁰(ℕ)=ℕ, 𝒥¹(ℕ)=𝒥(ℕ) and 𝒥ᵏ⁺¹(ℕ)=𝒥(𝒥ᵏ(ℕ)) for all k∈ℕ. The complexity of a numerical semigroup S is the minimun of the set {k∈ℕ|S ∈𝒥ᵏ(ℕ)}. In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.
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