Bracelet monoids and numerical semigroups

Abstract

Given positive integers $n_1,\ldots ,n_p$, we say that a submonoid M of $({\mathbb N},+)$ is a $(n_1,\ldots ,n_p)$-bracelet if $a+b+\left\{ n_1,\ldots ,n_p\right\} \subseteq M$ for every $a,b\in M\backslash \left\{ 0\right\}$. In this note, we explicitly describe the smallest $\left( n_1,\ldots ,n_p\right)$-bracelet that contains a finite subset X of ${\mathbb N}$. We also present a recursive method that enables us to construct the whole set $\mathcal B(n_1,\ldots ,n_p)=\left\{ M|M \quad \text {is a} \quad (n_1,\ldots ,n_p)\text {-bracelet}\right\}$. Finally, we study $(n_1,\ldots ,n_p)$-bracelets that cannot be expressed as the intersection of $(n_1,\ldots , n_p)$-bracelets properly containing it.