Numerical semigroups with concentration two

The first author was partially supported by MTM-2017-84890-P and by Junta de Andalucia group FQM-343. The second author is supported by the project FCT PTDC/MAT/73544/2006). 2010 Mathematics Subject Classification: 20M14, 11D07.We define the concentration of a numerical semigroup S as C(S) = max {next(S)(s) - s vertical bar s is an element of S\{0}} wherein next(S)(s) = min {x is an element of S vertical bar s < x}. In this paper, we study the class of numerical semigroups with concentration 2. We give algorithms to calculate the whole set of this class of semigroups with given multiplicity, genus or Frobenius number. Separately, we prove that this class of semigroups verifies the Wilf's conjecture.project FCT (Fundacao para a Ciencia e a Tecnologia) PTDC/MAT/73544/2006Junta de Andalucia FQM-343 MTM-2017-84890-

The multiples of a numerical semigroup

Given two numerical semigroups $S$ and $T$ we say that $T$ is a multiple of $S$ if there exists an integer $d \in \mathbb{N} \setminus \{0\}$ such that $S = \{x \in \mathbb{N} \mid d x \in T\}$. In this paper we study the family of multiples of a (fixed) numerical semigroup. We also address the open problem of finding numerical semigroups of embedding dimension $e$ without any quotient of embedding dimension less than $e$, and provide new families with this property.Comment: 15 page

Frobenius pseudo-varieties in numerical semigroups

This is a post-peer-review, pre-copyedit version of an article published in Annali di Matematica Pura ed Applicata. The final authenticated version is available online at: https://doi.org/10.1007/s10231-013-0375-1.The common behavior of several families of numerical semigroups led up to defining the Frobenius varieties. However, some interesting families were out of this definition. In order to overcome this situation, in this paper we introduce the concept of (Frobenius) pseudo-varieties. Moreover, we will show that most of the results for varieties can be generalized to pseudo-varieties.Both of the authors are supported by FQM-343 (Junta de Andalucía), MTM2010-15595 (MICINN, Spain), and FEDER funds. The second author is also partially supported by Junta de Andalucía/Feder grant number FQM-5849

The extended Frobenius problem for Lucas series incremented by a Lucas number

We study the extended Frobenius problem for sequences of the form $\{l_a\}\cup\{l_a+l_n\}_{n\in\mathbb{N}}$ and $\{l_a+l_n\}_{n\in\mathbb{N}}$, where $\{l_n\}_{n\in\mathbb{N}}$ is the Lucas series and $l_a$ is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf's conjecture.Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:2210.0081

Perfect numerical semigroups

A numerical semigroup is perfect if it does not have isolated gaps. In this paper we will order the perfect numerical semigroups with a fixed multiplicity. This ordering allows us to give an algorithm procedure to obtain them. We also study the perfect monoid, which is a subset of N that can be expressed as an intersection of perfect numerical semigroups, and we present the perfect monoid generated by a subset of N. We give an algorithm to calculate it. We study the perfect closure of a numerical semigroup, as well as the perfect numerical semigroup with maximal embedding dimension, in particular Arf and saturated numerical semigroups