The numerical semigroup of the integers which are bounded by a submonoid of N2 (original) (raw)

Numerical semigroups problem list

2013

A numerical semigroup is a subset of N (here N denotes the set of nonnegative integers) that is closed under addition, contains the zero element, and its complement in N is finite.

Associated semigroups of numerical sets with fixed Frobenius number

arXiv: Combinatorics, 2019

A numerical set is a co-finite Subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its compliment. A numerical semigroup is a numerical set that is closed under addition. Each numerical set has an associated semigroup A(T)=t∣t+TsubseteqTA(T)=\{t|t+T\subseteq T\}A(T)=t∣t+TsubseteqT, which is a numerical semigroup with the same Frobenius number as that of TTT. For a fixed Frobenius number fff there are 2f−12^{f-1}2f−1 numerical sets. We give a complete asymptotic description of what percentage of these numerical sets are mapped to which semigroups. We also obtain parallel results for symmetric numerical sets.

Numerical semigroups of small and large type

International Journal of Algebra and Computation, 2021

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 [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

Numerical semigroups with a given set of pseudo-Frobenius numbers

LMS Journal of Computation and Mathematics, 2016

The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of pseudo-Frobenius numbers of a numerical semigroup and, if so, to compute the set of all numerical semigroups having this set as set of pseudo-Frobenius numbers.

The covariety of numerical semigroups with fixed Frobenius number

arXiv (Cornell University), 2023

Denote by m(S) the multiplicity of a numerical semigroup S. A covariety is a nonempty family C of numerical semigroups that fulfills the following conditions: there is the minimum of C , the intersection of two elements of C is again an element of C and S\{m(S)} ∈ C for all S ∈ C such that S = min(C). In this work we describe an algorithmic procedure to compute all the elements of C. We prove that there exists the smallest element of C containing a set of positive integers. We show that A (F) = {S | S is a numerical semigroup with Frobenius number F } is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.

The genus, Frobenius number and pseudo-Frobenius numbers of numerical semigroups of type 2

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2016

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integersg1,g2, we give necessary and sufficient conditions in order to have a numerical semigroupSsuch that {g1,g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

Combinatory problems in numerical semigroups

2019

This thesis is devoted to the study of the theory of numerical semigroups. First, the focus is on saturated numerical semigroups. We will give algorithms that allows us to compute, for a given integer g (respectively integer F), the set of all saturated numerical semigroups with genus g (respectivaly with Frobenius number F). After that, we will solve the Frobenius problem for three particular classes of numerical semigroups: Mersenne, Thabit and Repunit numerical semigroups. Lastly, we will characterize and study the digital semigroups and the bracelet monoids; Resumo: Problemas Combinat´orios em Semigrupos Num´ericos Esta tese ´e dedicada ao estudo da teoria dos semigrupos num´ericos. O primeiro foco ´e o estudo dos semigrupos num´ericos saturados. Daremos algoritmos que nos ir˜ao permitir calcular, dado um inteiro g (repectivamente, um inteiro F), o conjunto de todos os semigrupos num´ericos saturados com g´enero g (respectivamente, com n´umero de Frobenius F). Depois disso, irem...

Numerical semigroups bounded by a cyclic monoid

Journal of Mathematical Inequalities, 2021

We will say that a numerical semigroup S is bounded by a cyclic monoid if there exist integer numbers 0 α < β such that S = {x ∈ N | kα < x < kβ for some k ∈ N} ∪ {0}. The goal of this work is to study this kind of numerical semigroups. In particular, we will determine important invariants of them such as multiplicity, embedding dimension, Frobenius number and genus.