J. C. Rosales

Irreducible Numerical Semigroups

Symmetric numerical semigroups are probably the numerical semigroups that have been most studied in the literature. The motivation and introduction of these semigroups is due mainly to Kunz, who in his manuscript [44] proves that a onedimensional analytically irreducible Noetherian local ring is Gorenstein if and only if its value semigroup is symmetric. Symmetric numerical semigroups always have odd Frobenius number. The translation of this concept for numerical semigroups with even Frobenius number motivates the definition of pseudo-symmetric numerical semigroups. In [5] it is shown that these semigroups also have their interpretation in one-dimensional local rings, since a numerical semigroup is pseudo-symmetric if and only if its semigroup ring is a Kunz ring.

Systems of Inequalities and Numerical Semigroups

A one-to-one correspondence is described between the setS(m) of numerical semigroups with multiplicity m and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in S(m) a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of Nm−1. Finally, this result is particularized to the symmetric case.

Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups Communicated by M.-F. Roy

Abstract We study those numerical semigroups that are intersections of symmetric numerical semigroups and we construct an algorithm to find this decomposition. These semigroups are characterized from their pseudo-Frobenius numbers.

Numerical semigroups with embedding dimension three

Abstract.Every numerical semigroup generated by three elements is determined by six positive integers that are the solution to a system of three polynomial equations. We give formulas of the Frobenius number and the cardinality of the set of gaps in terms of these six parameters.

ON PRESENTATIONS OF COMMUTATIVE MONOIDS

In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.

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.

Every Numerical Semigroup is One Half of Infinitely Many Symmetric Numerical Semigroups

Let S be a numerical semigroup. Then there exist infinitely many symmetric numerical semigroups such that . We give an explicit description of them.

Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143-172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.

A deterministic algorithm to decide if a finitely presented abelian monoid is cancellative

In this paper we give an algorithm to compute a finite presentation for any finitely generated commutative cancellative monoid, and in particular we apply it to derive an algorithm to decide whether a finitely presented commutative monoid is cancellative or not.

Proportionally modular diophantine inequalities and the Stern-Brocot tree

Given positive integers a, b and c to compute a generating system for the numerical semigroup whose elements are all positive integer solutions of the inequality ax mod b < cx is equivalent to computing a Bezout sequence connecting two reduced fractions. We prove that a proper Bezout sequence is completely determined by its ends and we give an algorithm to compute the unique proper Bezout sequence connecting two reduced fractions. We also relate Bezout sequences with paths in the Stern-Brocot tree and use this tree to compute the minimal positive integer solution of the above inequality.