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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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