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

Abstract

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 $a_1{x}_1+\cdots +a_{r-1}{x}_{r-1}=a_r{x}_r$, with $a_1,\dots ,a_r$ 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 Apéry sets with respect to the extremal rays.