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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
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