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Minimal systems of generators for ideals of semigroups. (English) Zbl 0913.20036
Let \(S\) be a commutative cancellative finitely generated (additive) semigroup with 0 such that \(S\cap(-S)=\{0\}\). Let \(\{n_1,\dots,n_r\}\subset S\) be a set of generators for \(S\). For a field \(k\), the semigroup algebra \(k[S]\) is an \(S\)-graded ring in a trivial way, and the polynomial ring \(R=k[X_1,\dots,X_r]\) can also be considered as an \(S\)-graded ring, assigning the degree \(n_i\) to \(X_i\), and the component \(R_m\) is a finite dimensional \(k\)-vector space (in fact the latter property is equivalent to the condition \(S\cap(-S)=\{0\}\)). Now the \(k\)-algebra epimorphism \(\phi\colon R\to k[S]\) defined by \(\phi(X_i)=n_i\) is a graded homomorphism of degree zero, hence its kernel \(I\) is a homogeneous ideal. This ideal is called “the defining ideal of the semigroup \(S\)”, or simply, by an abuse of language, “the ideal of \(S\)”. Based on the method of A. Campillo and P. Pisón [C. R. Acad. Sci., Paris, Sér. I 316, No. 12, 1303-1306 (1993; Zbl 0816.20050)] the present paper presents an algorithm to compute a minimal system of binomial generators for \(I\). Investigations of this kind have their origin in the study of algebraic curves and go back to J. Herzog [Manuscr. Math. 3, 175-193 (1970; Zbl 0211.33801)].

MSC:
20M05 Free semigroups, generators and relations, word problems
20M14 Commutative semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
16S36 Ordinary and skew polynomial rings and semigroup rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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