Numerical semigroups of maximal and almost maximal length
A numerical semigroup is a subsemigroup S of the natural numbers \({\mathbb{N}}\) with addition. Any numerical semigroup is finitely generated, and it will be assumed that the g.c.d. of the generators of S is one. Define the following three invariants of S: \(c(S)=| {\mathbb{N}}-S|\), \(g(S)=\max \{x\in {\mathbb{Z}}|\) \(x\not\in S\}\), \(t(S)=| S'|\), where \(S'=\{x\in {\mathbb{Z}}|\) \(x\not\in S\), \(x+s\in S\) for all \(s\in S- \{0\}\}\). These numbers correspond to numerical invariants of the complete local ring \(A=K[[t^{s_ 1},...,t^{s_ p}]]\), where \(S=<s_ 1,...,s_ p>\), K is any field, t an indeterminate over K; in fact, A is a one-dimensional Cohen-Macaulay local domain with Cohen- Macaulay type t(S); c(S) and r(S) are lengths of well-defined A-modules. For ring theoretical reasons, c(S)\(\leq r(S)t(S)\) must hold. The semigroup S is said to have maximal length if \(c(S)=r(S)t(S)\) and almost maximal length if \(c(S)=r(S)t(S)-1\). In the present paper all numerical semigroups of maximal and almost maximal length are determined. For the ring theoretical background see [\textit{R. Fröberg}, \textit{C. Gottlieb} and \textit{R. Häggkvist}, Semigroups, semigroup rings and analytically irreducible rings, Report, Univ. Stockholm, No.1 (1986)].
one-dimensional Cohen- Macaulay local domain
scientific article; zbMATH DE number 4190042