Inside factorial monoids and the cale monoid of a single Diophantine equation

We give a structure theorem for inside factorial domains. As an example we study the monoid of nonnegative integer solutions of equations of the form $a_1x_1+\cdots +a_{r-1}x_{r-1}=a_rx_r$, with $a_1,\ldots,a_r$ positive integers. This set is isomorphic to a simplicial full affine semigroup, and thus it can be described in terms of its extremal rays and the Ap\'ery sets with respect to the extremal rays.