Proportionally modular numerical semigroups 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 $ax \bmod {b} \le cx$ is said to be proportionally modular. For any interval $[\alpha ,\beta ]$, $S\big ([\alpha ,\beta ]\big )$ 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 $S\big ([\alpha ,\beta ]\big )$ equal S.