×

zbMATH — the first resource for mathematics

Indispensable binomials in semigroup ideals. (English) Zbl 1204.13014
The present paper deals with the problem of the uniqueness of a minimal set of binomial generators of a semigroup ideal. The main contribution of the paper is to give necessary and/or sufficient conditions for the uniqueness of such a minimal system of generators. The authors achieve these results by means of the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.
In the first part of the paper, the authors study a combinatorial description of indispensability, giving a combinatorial necessary and sufficient condition for the existence of indispensable binomials in a semigroup ideal (Theorem 8). In this part they also provide an explicit characterization of all indispensable binomials and monomials of a semigroup ideal (Corollary 11).
In the second part of the paper, the problem of the existence of indispensable binomials in a semigroup ideal is studied using Gröbner bases. Using these techniques the authors give in Theorem 13 effective necessary conditions for the existence of indispensable binomials.
The paper finishes with an example of application of the main results to a problem in Algebraic Statistics.

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
16W50 Graded rings and modules (associative rings and algebras)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Satoshi Aoki, Akimichi Takemura, and Ruriko Yoshida, Indispensable monomials of toric ideals and Markov bases, J. Symbolic Comput. 43 (2008), no. 6-7, 490 – 507. · Zbl 1170.13008 · doi:10.1016/j.jsc.2007.07.012 · doi.org
[2] Anna Maria Bigatti, Roberto La Scala, and Lorenzo Robbiano, Computing toric ideals, J. Symbolic Comput. 27 (1999), no. 4, 351 – 365. · Zbl 0958.13009 · doi:10.1006/jsco.1998.0256 · doi.org
[3] Emilio Briales, Pilar Pisón, Antonio Campillo, and Carlos Marijuán, Combinatorics of syzygies for semigroup algebras, Collect. Math. 49 (1998), no. 2-3, 239 – 256. Dedicated to the memory of Fernando Serrano. · Zbl 0929.13007
[4] E. Briales, A. Campillo, C. Marijuán, and P. Pisón, Minimal systems of generators for ideals of semigroups, J. Pure Appl. Algebra 124 (1998), no. 1-3, 7 – 30. · Zbl 0913.20036 · doi:10.1016/S0022-4049(96)00106-5 · doi.org
[5] Hara Charalambous, Anargyros Katsabekis, and Apostolos Thoma, Minimal systems of binomial generators and the indispensable complex of a toric ideal, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3443 – 3451. · Zbl 1127.13018
[6] H. Charalambous, A. Thoma, On simple \( \mathcal{A}\)-multigraded minimal resolutions, Contemporary Mathematics, Vol. 502, Amer. Math. Soc., 2009, pp. 33-44. · Zbl 1183.13017
[7] Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist. 26 (1998), no. 1, 363 – 397. · Zbl 0952.62088 · doi:10.1214/aos/1030563990 · doi.org
[8] David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1 – 45. · Zbl 0873.13021 · doi:10.1215/S0012-7094-96-08401-X · doi.org
[9] S. Eliahou, Courbes monomiales et algébre de Rees symbolique, PhD Thesis, Université of Genève, 1983.
[10] Dan Geiger, Christopher Meek, and Bernd Sturmfels, On the toric algebra of graphical models, Ann. Statist. 34 (2006), no. 3, 1463 – 1492. · Zbl 1104.60007 · doi:10.1214/009053606000000263 · doi.org
[11] Jürgen Herzog, Generators and relations of abelian semigroups and semigroup rings., Manuscripta Math. 3 (1970), 175 – 193. · Zbl 0211.33801 · doi:10.1007/BF01273309 · doi.org
[12] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. · Zbl 1090.13001
[13] Hidefumi Ohsugi and Takayuki Hibi, Indispensable binomials of finite graphs, J. Algebra Appl. 4 (2005), no. 4, 421 – 434. · Zbl 1093.13020 · doi:10.1142/S0219498805001265 · doi.org
[14] Hidefumi Ohsugi and Takayuki Hibi, Toric ideals arising from contingency tables, Commutative algebra and combinatorics, Ramanujan Math. Soc. Lect. Notes Ser., vol. 4, Ramanujan Math. Soc., Mysore, 2007, pp. 91 – 115. · Zbl 1187.13024
[15] I. Ojeda, A. Vigneron-Tenorio, Simplicial complexes and minimal free resolution of monomial algebras. J. Pure Appl. Algebra 214 (2010), no. 6, 850-861. · Zbl 1195.13015
[16] Ignacio Ojeda Martínez de Castilla and Pilar Pisón Casares, On the hull resolution of an affine monomial curve, J. Pure Appl. Algebra 192 (2004), no. 1-3, 53 – 67. · Zbl 1079.13007 · doi:10.1016/j.jpaa.2004.01.007 · doi.org
[17] Ignacio Ojeda Martínez de Castilla, Examples of generic lattice ideals of codimension 3, Comm. Algebra 36 (2008), no. 1, 279 – 287. · Zbl 1133.13014 · doi:10.1080/00927870701665487 · doi.org
[18] Irena Peeva and Bernd Sturmfels, Generic lattice ideals, J. Amer. Math. Soc. 11 (1998), no. 2, 363 – 373. · Zbl 0905.13005
[19] Irena Peeva and Bernd Sturmfels, Syzygies of codimension 2 lattice ideals, Math. Z. 229 (1998), no. 1, 163 – 194. · Zbl 0918.13006 · doi:10.1007/PL00004645 · doi.org
[20] Pilar Pisón-Casares and Alberto Vigneron-Tenorio, On Lawrence semigroups, J. Symbolic Comput. 43 (2008), no. 11, 804 – 810. · Zbl 1166.13028 · doi:10.1016/j.jsc.2008.02.003 · doi.org
[21] J. C. Rosales and P. A. García-Sánchez, Finitely generated commutative monoids, Nova Science Publishers, Inc., Commack, NY, 1999. · Zbl 0966.20028
[22] Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. · Zbl 0856.13020
[23] Akimichi Takemura and Satoshi Aoki, Some characterizations of minimal Markov basis for sampling from discrete conditional distributions, Ann. Inst. Statist. Math. 56 (2004), no. 1, 1 – 17. · Zbl 1049.62068 · doi:10.1007/BF02530522 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.