On proportionally modular numerical semigroups that are generated by arithmetic progressions

Abstract

A numerical semigroup is a submonoid of ${\mathbb{Z}}_{\ge 0}$ whose complement in ${\mathbb{Z}}_{\ge 0}$ is finite. For any set of positive integers $a,b,c$, the numerical semigroup $S(a,b,c)$ formed by the set of solutions of the inequality $ax\mathrm{mod}b\le cx$ is said to be proportionally modular. For any interval $[\alpha ,\beta ]$, $S([\alpha ,\beta ])$ is the submonoid of ${\mathbb{Z}}_{\ge 0}$ obtained by intersecting the submonoid of ${\mathbb{Q}}_{\ge 0}$ generated by $[\alpha ,\beta ]$ with ${\mathbb{Z}}_{\ge 0}$. For the numerical semigroup $S$ generated by a given arithmetic progression, we characterize $a,b,c$ and $\alpha ,\beta$ such that both $S(a,b,c)$ and $S([\alpha ,\beta ])$ equal $S$.

15 pages, 2 figures