Generators and relations of abelian semigroups and semigroup rings

Abstract

The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Apéry [1]. This result applies to algebraic geometry: An affine space-curve C with the parametric equations x=t^a, y=t^b, z=t^c, a, b, c natural numbers with greatest common divisor 1, is a global idealtheoretic complete intersection, if and only if the semigroup S generated by a, b, c is symmetric.