Irreducible numerical semigroups

Abstract

This is a survey of several results obtained with M. B. Branco at the Universidade deÉvora, J. I. García-García at the Universidad de Cádiz, J. A. Jiménez-Madrid at the Instituto de Ciencias Matemáticas, and J. C. Rosales at the Universidad de Granada. A numerical semigroup is a submonoid of the monoid of nonnegative integers (under addition) with finite complement in it. A numerical semigroup is irreducible if it cannot be expressed as the intersection of two numerical semigroups properly containing it. Every numerical semigroup can be written as a (finite) intersection of irreducible numerical semigroups. Minimal decompositions can be thought in terms of cardinality or redundancy. Both concepts do not coincide, and uniqueness is far from being reached. As we have mentioned above, the complement of a numerical semigroup in the set of nonnegative integers is finite. So it makes sense to think about the largest integer not belonging to a numerical semigroup. This integer is known as the Frobenius number, which became really famous mainly for the lack of a general (algebraic) formula to calculate it for numerical semigroups with more than two minimal generators. Irreducible numerical semigroups can be also characterized as those numerical semigroups maximal (with respect to set inclusion) in the set of numerical semigroups with a fixed Frobenius number. This allows to relate this concept with two well known families of numerical semigroups: symmetric and pseudo-symmetric numerical semigroups. An irreducible numerical semigroup is either symmetric or pseudo-symmetric (depending on the parity of its Frobenius number), and the union of the set of symmetric and the set of pseudo-symmetric numerical semigroups yields that of irreducible numerical semigroups. Symmetric numerical semigroups earned some popularity due to a result by Kunz that states that the semigroup ring associated to a numerical semigroup is Gorenstein if and only if the numerical semigroup is symmetric (semigroup rings associated to pseudo-symmetric numerical semigroups are Kunz rings). Several subfamilies of the class of symmetric numerical semigroups were also widely studied due to their relevance as examples in Algebraic Geometry. These include that of complete intersections, free (in the sense of Bertin and Carbonne) and telescopic numerical semigroups. In the set of numerical semigroups with a given Frobenius number, the average of irreducible numerical semigroups is really poor. However, this apparent lack contrasts not

On a conjecture by Wilf about the Frobenius number

“…In other words, η j counts the intervals I k containing exactly j elements of S, while ǫ j only counts such intervals among the first Q. The two definitions differ slightly, and the numbers η j can be expressed in terms of Ap(S): 12,13). The following properties hold:…”

Classes of complete intersection numerical semigroups

“…A semigroup S is called a complete intersection if the semigroup ring k[[t S ]] is complete intersection, or equivalently if the cardinality of any of its minimal presentations equals ν(S) − 1 (cf. [27], page 129).…”

“…, ν we get (γ σ(i) + 1)n i = (γ σ(i) + 1)g σ(i) = ν j=1 λ σ(i),j g j = ν j=1 λ σ(i),σ(j) g σ(j) = ν j=1 λ σ(i),σ(j) n j thus φ i ≤ γ σ(i) by the triangularity of the matrix L σ . Following the notation of [27], let…”

The first elements of the quotient of a numerical semigroup by a positive integer

“…There is a strict relation (cf. [3], [5], [6]) between this problem and the problem of finding an explicit formula for the smallest positive integer K such that Ka 3 is representable as Ka 3 = λ 1 a 1 + λ 2 a 2 , where a 1 , a 2 , a 3 are pairwise coprime positive integers and λ 1 , λ 2 ∈ N. In this short paper we approach this problem by studying it in the context of quotients of numerical semigroups (cf. [6] for a good monograph on numerical semigroups).…”

“…[3], [5], [6]) between this problem and the problem of finding an explicit formula for the smallest positive integer K such that Ka 3 is representable as Ka 3 = λ 1 a 1 + λ 2 a 2 , where a 1 , a 2 , a 3 are pairwise coprime positive integers and λ 1 , λ 2 ∈ N. In this short paper we approach this problem by studying it in the context of quotients of numerical semigroups (cf. [6] for a good monograph on numerical semigroups). In Section 1 we study the set that contains the first elements of a quotient of a numerical semigroup generated by two integers, by explaining the relation between the three quotients a i ,a j a k that can be made with the same triple a 1 , a 2 , a 3 : we find that, once the first elements of one of those three quotients is known, it is possible to deduce the first elements of the other two.…”