On Symmetric Numerical Semigroups

A numerical semigroup is a subset S of N such that is closed under sums, contains the zero and generates Z as a group. From this definition Ž w x. Ž one obtains see, for example, 2 that S has a conductor C i.e., the . maximum among all the natural numbers not belonging to S . A numerical semigroup S is called symmetric if for every integer z f S, C y z g S. The study of the subsemigroups of N is a classical problem, which is equivalent to study of the sets of natural solutions of linear equations with w x coefficients in N. From 1970 and after the works 5, 6 and others, the study of the subsemigroups of N is also motivated by its applications to algebraic geometry. An example arises when we consider an analytically irreducible one-dimensional Noetherian local domain R, the integral closure V of R in the quotient field of R is a discrete valuation ring, if we assume that R and V have the same residue class field and we denote by Ž . ̈ the valuation attached to V, then S s ̈ R is a numerical semigroup. R Then properties on the semigroup S imply properties of the ring R. For R w x Ž w x. example, 6 proves that R is Gorenstein see 1 if and only if S is R symmetric. The close relation between numerical semigroups and monomial curves has produced the usage of algebraic geometry terminology in the field of numerical semigroups. For example, the smallest integer m greater than zero in S is usually called the multiplicity of S, the cardinal e of a minimal system of generators for S is called the embedding dimension of S, and a finite presentation for S is called a relation for S.