Ignacio Ojeda

Uniquely presented finitely generated commutative monoids

A finitely generated commutative monoid is uniquely presented if it has a unique minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally, for some relevant families of numerical semigroups we describe the elements that are uniquely presented.

Indispensable binomials in semigroup ideals

In this paper, we deal with the problem of the uniqueness of a minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for the uniqueness of such a minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.

FACTORIZATION INVARIANTS IN HALF-FACTORIAL AFFINE SEMIGROUPS

Let be the monoid generated by We introduce the homogeneous catenary degree of as the smallest N ∈ ℕ with the following property: for each and any two factorizations u, v of a, there exist factorizations u = w1,…,wt = v of a such that, for every k, d(wk,wk+1) ≤ N, where d is the usual distance between factorizations, and the length of wk, |wk|, is less than or equal to max{|u|, |v|}. We prove that the homogeneous catenary degree of improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the ω-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.

Simplicial complexes and minimal free resolution of monomial algebras

Abstract This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. We explicitly describe how the graded minimal free resolution of those algebras is related to the combinatorics of some simplicial complexes. Our description may be interpreted as an algorithmic procedure to partially compute this resolution.

AFFINE SEMIGROUPS HAVING A UNIQUE BETTI ELEMENT

We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups. As an example, we particularize our description to numerical semigroups.

Binomial Canonical Decompositions of Binomial Ideals

In this article, we prove that every binomial ideal in a polynomial ring over an algebraically closed field of characteristic zero admits a canonical primary decomposition into binomial ideals. Moreover, we prove that this special decomposition is obtained from a cellular decomposition which is also defined in a canonical way and does not depend on the field.

On the computation of the Apéry set of numerical monoids and affine semigroups

A simple way of computing the Apéry set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

Kronecker Square Roots and the Block Vec Matrix

Abstract Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a matrix has a square root for the Kronecker product.

Index of nilpotency of binomial ideals

Abstract In this paper, we study and compute bounds for the index of nilpotency of lattice and cellular binomial ideals which are optimal in many cases. This computations can be generalized to binomial ideals, getting an effective Hilberts Nullstellensatz for binomial ideals.

The short resolution of a semigroup algebra

This work generalizes the short resolution given in Proc. Amer. Math. Soc. \textbf{131}, 4, (2003), 1081--1091, to any affine semigroup. Moreover, a characterization of Apery sets is given. This characterization lets compute Apery sets of affine semigroups and the Frobenius number of a numerical semigroup in a simple way. We also exhibit a new characterization of the Cohen-Macaulay property for simplicial affine semigroups.