Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension (Q2718989)

A numerical semigroup is an additive submonoid of \(\mathbb{N}\) which generates \(\mathbb{Z}\) as a group. For a numerical semigroup \(S\), its Frobenius number, denoted by \(C(S)\), is the greatest integer not in \(S\); and \(S\) is called symmetric if \(z\in\mathbb{Z}-S\) implies \(C(S)-z\in S\). Also, \(S\) has a unique set of minimal generators \(\{n_0<n_1<\cdots<n_p\}\). The integers \(n_0\) and \(p+1\) are called the multiplicity and embedding dimension of \(S\), and are denoted by \(m(S)\) and \(\mu(S)\), respectively. It is known that if \(S\) is a symmetric numerical semigroup and \(m(S)\geq 3\), then \(\mu(S)\leq m(S)-1\). In this paper, for any integers \(m\) and \(e\) such that \(2\leq e\leq m-1\), the author constructs a symmetric numerical semigroup \(S\) with \(m(S)=m\) and \(\mu(S)=e\). He also shows that if \(\mu(S)\geq 3\), then the cardinality of any minimal presentation for such semigroups is \(\mu(S)(\mu(S)-1)/2-1\).