Union of sets of lengths of numerical semigroups

J. I. García-García; D. Marín-Aragón; A. Vigneron-Tenorio

paper

Union of sets of lengths of numerical semigroups

arXiv:1906.01266

Abstract

Let $S = ⟨ a_{1}, \dots, a_{p} ⟩$ be a numerical semigroup, $s \in S$ and $z (s)$ its set of factorizations. The set of length is denoted by $L (s) = {L (x_{1}, \dots, x_{p}) ∣ (x_{1}, \dots, x_{p}) \in Z (s)}$ where $L (x_{1}, \dots, x_{p}) = x_{1} + \dots + x_{p}$ . From these definitions, the following sets can be defined $W (n) = {s \in S ∣ \exists x \in z (s) such that L (x) = n}$ , $ν (n) = \cup_{s \in W (n)} L (s) = {l_{1} < l_{2} < \dots < l_{r}}$ and $Δ ν (n) = {l_{2} - l_{1}, \dots, l_{r} - l_{r - 1}}$ . In this paper, we prove that the set $Δ ν (S) = \cup_{n \in N} Δ ν (n)$ is almost periodic with period $lcm (a_{1}, a_{p})$ .