Numerical Semigroups Research Articles

On a class of numerical semigroups with embedding dimension equal to 4

We give a description of a class of numerical semigroups with embedding dimension equal to 4, defined by four pairwise relatively prime nonnegative integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] such that [Formula: see text], for [Formula: see text] and [Formula: see text]. Such description provides a mode to determine the characteristics of the corresponding numerical semigroups: the Frobenius number, gaps, genus, etc.

Counting the numerical semigroups with a specific special gap

Let S be a numerical semigroup. An element is a special gap of S if is also a numerical semigroup. If a is a positive integer, we denote by the set of all numerical semigroups for which a is a special gap. We say that an element of is -irreducible if it cannot be expressed as the intersection of two numerical semigroups of properly containing it. The main aim of this paper is to describe three algorithmic procedures: the first one calculates the elements of the second one determines whether or not a numerical semigroup is -irreducible and the third one computes all the -irreducibles numerical semigroups.

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.

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 Corner Element of Generalized Numerical Semigroups

In this paper we introduce the concept of corner element of a generalized numerical semigroup, which extends in a sense the idea of conductor of a numerical semigroup to generalized numerical semigroups in higher dimensions. We present properties of this new notion and its relations with existing invariants in the literature, and provide an algorithm to compute all the generalized numerical semigroups with fixed corner. Besides that, we provide lower and upper bounds on the number of generalized numerical semigroups having a fixed corner element.

On the occurrence of complete intersections in shifted families of numerical semigroups

On the occurrence of complete intersections in shifted families of numerical semigroups

Counting the ideals with given genus of a numerical semigroup

If [Formula: see text] is a numerical semigroup, denote by [Formula: see text] the genus of [Formula: see text]. A numerical semigroup [Formula: see text] is an [Formula: see text]-semigroup if [Formula: see text] is an ideal of [Formula: see text]. If [Formula: see text], then we denote by [Formula: see text] the number of [Formula: see text]-semigroups with genus [Formula: see text]. In this work, we conjecture that [Formula: see text] if [Formula: see text] and we show that there is a term from which this sequence becomes stationary. That is, there exists [Formula: see text] such that [Formula: see text] [Formula: see text] for all [Formula: see text] Moreover, we prove that the conjecture is true for ordinary numerical semigroups, that is, numerical semigroups which the form [Formula: see text] for some positive integer. Additionally, we calculate the term from which the sequence becomes stationary.

The examination of the quotient of numerical semigroup with RF-matrices

In this paper, we study quotients of a numerical semigroups with RF (Row-Factorization) matrices. We prove a formula for the Frobenious number of quotients of some families of numerical semigroups. Moreover, we examine half of the numerical semigroups, pseudo-symmetric numerical semigroups.

About Some Numerical SemigroupsWith Embedding Dimension Three

we will examine a family of numerical semigroups embedding dimension three, such that S_u=&lt; 8,8u+2,8u+9&gt; where u &gt;=1, u is integer . Also, we will give some relationsbetween these semigroups and their Arf closure.

Supersymmetric gaps of a numerical semigroup with two generators

In this paper we introduce the new concepts of supersymmetric and self-symmetric gaps of a numerical semigroup with two generators. Those concepts are based on certain symmetries of the gaps of the semigroup with respect to their Wilf number. We prove that the set of supersymmetric and self-symmetric gaps completely determines the semigroup and we compare this set with the fundamental gaps of the semigroup.

Frobenius R-Variety of the Numerical Semigroups Contained in a Given One

Frobenius R-Variety of the Numerical Semigroups Contained in a Given One

Gömme Boyutu Üç Olan Bazı Pseudo-Simetrik Sayısal Yarıgruplarının Delta Kümeleri Üzerine

Let S be a numerical semigroup. The catenary degree of an element s in S is a non-negative integer used to measure the distance between factorizations of s. The catenary degree of the numerical semigroup S is obtained at the maximum catenary degree of its elements. The maximum catenary degree of S is attained via Betti elements of S with complex properties. The Betti elements of S can be obtained from all minimal presentations of S. A presentation for S is a system of generators of the kernel congruence of the special factorization homomorphism. A presentation is minimal if it can not be converted to another presentation, that is, any of its proper subsets is no longer a presentation. The Delta set of S is a factorization invariant measuring the complexity of sets of the factorization lengths for the elements in S.&#x0D; In this study, we will mainly express the given above invariants of a special pseudo-symmetric numerical semigroup family in terms of its generators.

A New Face Iterator for Polyhedra and for More General Finite Locally Branched Lattices

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, García-Sánchez, O’Neill, and Wilburne, we apply this algorithm to prove Wilf’s conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf’s conjecture.

Positioned Numerical Semigroups with Small Gender

Positioned Numerical Semigroups with Small Gender

Numerical semigroups generated by quadratic sequences

We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Apéry set, as well as bounds on the elements of the Apéry set. We also find bounds on the Frobenius number and genus, and the asymptotic behavior of the Frobenius number and genus. Finally, we find the embedding dimension of all such numerical semigroups.

Type and conductor of simplicial affine semigroups

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K [ S ] . We define the type of S , type ( S ) , in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K [ S ] is Cohen-Macaulay. If K [ S ] is a d -dimensional Cohen-Macaulay ring of embedding dimension at most d + 2 , then type ( S ) ≤ 2 . Otherwise, type ( S ) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.

Positioned numerical semigroups with maximal gender as function of multiplicity and Frobenius number

A $C$-semigroup (respectively a $D$-semigroup) is a positioned numerical semigroup $S$ such that $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-1}{2}$ (respectively $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-2}{2}$). In this paper we study these semigroups giving formulas for the Frobenius number, pseudo-Frobenius number, and type. Furthermore, we give algorithms for computing the whole set of $C$-semigroups and $D$-semigroups.

Cyclotomic exponent sequences of numerical semigroups

We study the cyclotomic exponent sequence of a numerical semigroup S, and we compute its values at the gaps of S, the elements of S with unique representations in terms of minimal generators, and the Betti elements b∈S for which the set {a∈Betti(S):a≤Sb} is totally ordered with respect to ≤S (we write a≤Sb whenever a−b∈S, with a,b∈S). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection.

A Complete Classification of Planar Graphs Associated with the Ideal of the Numerical Semigroup

A Complete Classification of Planar Graphs Associated with the Ideal of the Numerical Semigroup

Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups

For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many previous results, provides more natural and intuitive proofs, and yields a completely explicit error bound.