On symmetric numerical semigroups. (English) Zbl 0856.20043

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 \(m\) is the smallest positive integer in \(S\), its embedding dimension \(e\) is the number of minimal generators of \(S\), and \(S\) is symmetric if \(C-z\in S\) for every integer \(z\notin S\). In this paper, the author gives an upper bound on the cardinality of a minimal relation for a symmetric numerical semigroup, investigates the symmetric numerical semigroups with \(3\leq e=m-1\), and studies the set \({\mathcal S}(m,c)\) of all numerical semigroups with conductor \(C\) and multiplicity \(m\).


20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems
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