System proportionally modular numerical semigroups

Given $a$, $b$ and $c$ integers with $b\not=0$, the set of integer solutions to the inequality $ax\mod b\leq cx$ is a numerical semigroup. Numerical semigroups of this form are called proportionally modular. A numerical semigroup is system proportionlally modular if it is the intersection of proportionally modular numerical semigroups. Not every numerical semigroup is of this form. We show how to decide whether or not a numerical semigroup is of this type. We also give a procedure to construct recurrently the set of all system proportionally modular numerical semigroups and show that these semigroups correspond with those that can be expressed as intersection of numerical semigroups of the form $\frac{\langle a,b\rangle}L$, with $a,b,L$ relatively prime positive integers. Toms proved that for these semigroups there exists a simple, separable, amenable and unital $C^*$-algebra with ordered $K_0$-group isomorphic to ${\mathbb Z}$ with positive cone $S$.