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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)].

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)
Full Text: DOI
[1] Azevedo, A., The Jacobian ideal of a plane algebroid curve, ()
[2] Bertin, J.; Carbonne, P., Semi-groups d’entiers et applications aux branches, J. algebra, 49, 81-95, (1977) · Zbl 0498.14016
[3] Bresinsky, H., Binomial generating sets for monomial curves, with applications of · Zbl 0738.14017
[4] Briales, E.; Campillo, A.; Marijuán, C.; Pisón, P., Combinatoric and syzygias of semigroup algebras, (1995), preprint
[5] Bruns, W.; Herzog, J., Cohen-Macaulay rings, (1993), Cambridge University Press · Zbl 0788.13005
[6] Campillo, A.; Marijuán, C., Higher relations for a numerical semigroup, Sem. theorie des nombres, Bordeaux, 3, 249-260, (1991) · Zbl 0818.20078
[7] Campillo, A.; Pisón, P., Generators of a monomial curve and graphs for the associated semigroup, Bull. soc. math. belgique, V. serie A, 45, 1-2, 45-58, (1993) · Zbl 0802.14014
[8] Campillo, A.; Pisón, P., L’idéald’un semi-grupe de type fini, C.R. acad. sci. Paris, Série I, 316, 1303-1306, (1993) · Zbl 0816.20050
[9] Cohen, H., A course in computational algebraic number theory, () · Zbl 0786.11071
[10] Eliahou, S., Courbes monomiales et algébre de Rees symbolique, ()
[11] Herzog, J., Generators and relations of semigroups and semigroup rings, Manuscripta math., 3, 175-193, (1970) · Zbl 0211.33801
[12] Herzog, J.; Kunz, E., Die wertehalbgruppe eines lokalen rings der dimension 1, Sitz. ber. heidelberger akad. d. wissengeh., 2, 27-67, (1971) · Zbl 0212.06102
[13] Kamoi, Y., Defining ideals of Cohen-Macaulay semigroups rings, Comm. algebra, 20, 3163-3189, (1992) · Zbl 0815.13013
[14] Kunz, E., (), 1-81, Preprint Fak. für Math. der Universität Regensburg
[15] Marijuán, C., Grafos asociados a un semigrupo numérico, (), 54
[16] Pisón, P., Métodos combinatorios en algebra local y curvas monomiales en dimensión 4, ()
[17] Pisón, P., El ideal de un semigrupo finitamente generado, ()
[18] Pisón, P.; Briales, E.; Borrego, J.; Pérez, M., Computing with ideals of semigroups, (1993), Preprint Universidad de Sevilla
[19] Prufer, H., Neuer beweis eines satzes ùber permutationen, Arch. der math. phys., 27, 3, 142-144, (1918) · JFM 46.0106.04
[20] J.C. Rosales, An algorithmic method to compute a minimal relation for any numerical semigroup, Internat. J. Algebra Comput., to appear. · Zbl 0863.20026
[21] Shrijver, A., Theory of linear and integer programming, (1986), Wiley-Interscience
[22] Sturmfels, B., Grobner bases and convex polytopes, (1995), American Mathematical Society New York
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