Narrow Results

Let ℕ be the set of nonnegative integers and let I be an interval of positive rational numbers. Then is a numerical semigroup. In this paper we study the multiplicity of the numerical semigroups of the form , where a and b are integers such that 2 ≤ a < b. We also see the connection between the multiplicity and the Frobenius number of this type of semigroups.

In this paper we introduce the notion of m-irreducibility that extends the standard concept of irreducibility of a numerical semigroup when the multiplicity is fixed. We analyze the structure of the set of m-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroup with multiplicity m into m-irreducible numerical semigroups.

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ ℕ | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is , where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

In this work, we give parametrizations in terms of the Kunz coordinates of numerical semigroups with multiplicity up to $5$. We also obtain parametrizations of MED semigroups, symmetric and pseudo-symmetric numerical semigroups with multiplicity up to $5$. These parametrizations also lead to formulas for the number of numerical semigroups, the number of MED semigroups and the number of symmetric and pseudo-symmetric numerical semigroups with multiplicity up to $5$ and given conductor.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number $F$, genus $g$ and type $t$. It is known that for any numerical semigroup $\frac{g}{F+1-g}\le t\le 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. The number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$.

In this work, we study $A$-semigroups, that is, numerical semigroups which have no consecutive elements less than the Frobenius number. We give algorithms that allow computation of the whole set of $A$-semigroups with a given genus, multiplicity and Frobenius number and from this we study interesting families of $A$-semigroups which are Frobenius varieties, pseudo-varieties; R-varieties.

We give a formula for the Frobenius vector of a free affine simplicial semigroup. This generalizes to the affine case a well known formula for free numerical semigroups.

We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to 7.

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps].

We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviors.

If $S$ is a numerical semigroup, denote by $\mathrm{\text{g}}(S)$ the genus of $S$. A numerical semigroup $T$ is an $\mathrm{\text{I}}(S)$-semigroup if $T\setminus \{0\}$ is an ideal of $S$. If $k\in \mathbb{N}$, then we denote by $\mathrm{\text{i}}(S,k)$ the number of $\mathrm{\text{I}}(S)$-semigroups with genus $\mathrm{\text{g}}(S)+k$. In this work, we conjecture that $\mathrm{\text{i}}(S,a)\le \mathrm{\text{i}}(S,b)$ if $a\le b,$ and we show that there is a term from which this sequence becomes stationary. That is, there exists $k_S\in \mathbb{N}$ such that $\mathrm{\text{i}}(S,k_S)=$$\mathrm{\text{i}}(S,k_S+h)$ for all $h\in \mathbb{N}.$ Moreover, we prove that the conjecture is true for ordinary numerical semigroups, that is, numerical semigroups which the form $\{0,m,\to \}$ for some positive integer. Additionally, we calculate the term from which the sequence becomes stationary.

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by a shifted geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Apéry set elements.

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation $a_1{x}_1+\cdots +a_k{x}_k=n$ ($a_1,\dots ,a_k$ are given positive integers with $gcd(a_1,\dots ,a_k)=1$) does not have a non-negative integer solution $(x_1,\dots ,x_k)$. The generalized Frobenius number (called the $p$-Frobenius number) is the largest integer such that this linear equation has at most $p$ solutions. That is, when $p=0$, the $0$-Frobenius number is the original Frobenius number.

In this paper, we introduce and discuss $p$-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer $p$, $p$-gaps, $p$-symmetric semigroups, $p$-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When $p=0$, they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

Given two numerical semigroups S and S', the distance between S and S' is the cardinality of S\S' plus the cardinality of S'\S. In this paper we study those numerical semigroups S for which there is a symmetric numerical semigroup whose distance to S is one.

We introduce a method to find upper and lower bounds for the genus of numerical semigroups. Using it we prove some old and new bounds for it and for the Frobenius number of the semigroup.

Given relatively prime integers a₁,…, a_n, the Frobenius number g(a₁,…, a_n) is defined as the largest integer which cannot be expressed as x₁a₁ +⋯+ x_na_n with x_i nonnegative integers. In this paper, we give the Frobenius number of primitive Pythagorean triples:

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n², (n + 1)², …} is asymptotically O(n²), verifying thus a conjecture of Dutch and Rickett. Furthermore, we ask a question on the representation of integers as sum of four large squares.

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.

Let $k,n$ be two positive integers such that $2\le k\le 2^n$ and $S(k,n)$ the numerical semigroup generated by $\{(2^k-1)\cdot 2^{n+i}-1|i\in \mathbb{N}\}$. Then $S(2,n)$ is the Thabit numerical semigroup introduced by J. C. Rosales, M. B. Branco and D. Torrão. In this paper, we give formulas for computing the Frobenius number, the genus and the embedding dimension of $S(k,n)$.

We give a description of a class of numerical semigroups with embedding dimension equal to 4, defined by four pairwise relatively prime nonnegative integers $n$, $x$, $y_1$ and $y_2$ such that $y_1+y_2=tn$, for $t\ge 2$ and $t\in \mathbb{N}$. Such description provides a mode to determine the characteristics of the corresponding numerical semigroups: the Frobenius number, gaps, genus, etc.