Dense numerical semigroups (Q2037093)

Let \(\mathbb{N}\) denote the set of nonnegative integers. A numerical semigroup is a submonoid \(S\) of \(\mathbb{N}\) under addition such that \(\mathbb{N}\setminus S\) has finitely many elements (known as gaps). The genus of \(S\) is the cardinality of \(\mathbb{N}\setminus S\), and the Frobenius number of \(S\) is the largest integer not in \(S\). The multiplicity of \(S\) is the least positive integer in \(S\). Every numerical semigroup is atomic (every element is the sum of finitely many atoms or irreducibles). The set of atoms of a numerical semigroup \(S\) is precisely the unique minimal generating set of the semigroup, \(((S\setminus\{0\})+(S\setminus\{0\}))\setminus(S\setminus\{0\})\), and its cardinality is known as the embedding dimension of \(S\), which is always smaller than or equal to the multiplicity of \(S\). A numerical semigroup \(S\) is called dense by the authors if for every \(s\in S\), the set \(S\cap\{s-1,s+1\}\) is not empty. The authors provide a series of procedures to compute the set of all dense numerical semigroups with fixed Frobenius number, and with fixed Frobenius number and genus. It is easy to check that there are only finitely many dense numerical semigroups with a given multiplicity; the authors describe a way to compute this set. Also, a method to compute the set of all dense numerical semigroups with fixed multiplicity, genus and Frobenius number is provided. The authors also present a way to compute the set of all dense oversemigroups of a given numerical semigroup (numerical semigroups containing the given semigroup). The last section gives a description of all dense numerical semigroups with embedding dimension three, and for these an explicit construction of the Apéry set with respect to the multiplicity is given (elements \(s\) in the semigroup such that \(s\) minus the multiplicity is no longer in the semigroup). As a consequence, with the help of Selmer's formulas, the authors present formulas for the Frobenius number and genus of any dense numerical semigroup with embedding dimension three.