On numerical semigroups. (English) Zbl 0853.20041

A numerical semigroup \(S\) is an additive submonoid of \(\mathbb{N}\) such that \(\mathbb{N}-S\) is finite. For a numerical semigroup \(S\), its conductor \(C\) is the largest integer not in \(S\), its multiplicity is the smallest positive integer \(m\) in \(S\), and the embedding dimension of \(S\) is the number of minimal generators of \(S\). In this paper, the author investigates the relationship between the two numerical semigroups \(S\) and \(S'=S\cup\{C\}\), and uses this to study the set \(S(m)\) of all numerical semigroups with multiplicity \(m\). The author also studies those numerical semigroups \(S\) with multiplicity equal to embedding dimension, i.e., with maximal embedding dimension in \(S(m)\).


20M14 Commutative semigroups
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