Set Of Numerical Semigroups Research Articles

Arithmetic Varieties of Numerical Semigroups

In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the rooted tree associated with an arithmetic variety. This tree is not locally finite; however, if the Frobenius number is fixed, the tree has finitely many nodes and algorithms can be developed. All algorithms provided in this article include their (non-debugged) implementation in GAP.

Distribution of genus among numerical semigroups with fixed Frobenius number

A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number f and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number f have genus close to frac{3f}{4}. We denote the number of numerical semigroups of Frobenius number f by N(f). While N(f) is not monotonic we prove that N(f)<N(f+2) for every f.

The weak Lefschetz property of graded Gorenstein algebras associated to the Apéry set of a numerical semigroup

It has been conjectured that all graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apéry set of M-pure symmetric numerical semigroups generated by four natural numbers. These algebras are graded Artinian Gorenstein algebras of codimension three.

Semigroups with fixed multiplicity and embedding dimension

Given m ∈ ℕ , a numerical semigroup with multiplicity m is called a packed numerical semigroup if its minimal generating set is included in { m , m + 1, …, 2 m − 1} . In this work, packed numerical semigroups are used to build the set of numerical semigroups with a given multiplicity and embedding dimension, and to create a partition of this set. Wilf’s conjecture is verified in the tree associated to some packed numerical semigroups. Furthermore, given two positive integers m and e , some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. We also compute the semigroups where these minimal values are achieved.

The Frobenius number in the set of numerical semigroups with fixed multiplicity and genus

We compute all possible numbers that are the Frobenius number of a numerical semigroup when multiplicity and genus are fixed. Moreover, we construct explicitly numerical semigroups in each case.

Two-Extension of a Numerical Semigroup with Embedding Dimension Two

Let S and be two numerical semigroups such that , and let d be a positive integer. We say that is a d-extension of S if . In this paper we study the set of numerical semigroups S such that S is a 2-extension of ⟨ n 1, n 2 ⟩.

Representation of Numerical Semigroups by Dyck Paths

We introduce square diagrams that represent numerical semigroups and we obtain an injection from the set of numerical semigroups into the set of Dyck paths.

Patterns on numerical semigroups

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabilizing ascending chain whose first step consists just of the trivial semigroup and whose second step is the well-known class of Arf semigroups. We describe a procedure to compute the closure of a numerical semigroup with respect to a pattern. By using the concept of system of generators associated to a pattern, we construct recursively a directed acyclic graph with all the semigroups admitting the pattern.

Numerical Semigroups with a Monotonic Apery Set

We study numerical semigroups S with the property that if m is the multiplicity of S and w(i) is the least element of S congruent with i modulo m, then 0 < w(1) < ... < w(m − 1). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups.