On numerical semigroups with high embedding dimension (Q1266467)

Let \(S\) be a numerical semigroup, that is, an additive submonoid of \(\mathbb{N}\) which generates \(\mathbb{Z}\) as a group. Let \(\{n_0<n_1<\cdots<n_p\}\) be a minimal system of generators for \(S\), let \(\varphi\colon\mathbb{N}^{p+1}\to\mathbb{N}\) be the semigroup homomorphism \(\varphi(a_0,a_1,\dots,a_p)=a_0n_0+a_1n_1+\cdots+a_pn_p\), and let \(\sigma\) be the kernel congruence of \(\varphi\). In this paper, the authors continue the investigation of the first-named author on the number of elements in a system of generators for \(\sigma\) with minimal cardinality. Specifically, they study the cases \(p=n_0-2\) and \(p=n_0-3\) for arbitrary numerical semigroups and the cases \(p=n_0-3\) and \(p=n_0-4\) for symmetric numerical semigroups.