On divisor-closed submonoids and minimal distances in finitely generated monoids

We study the lattice of divisor-closed submonoids of finitely generated cancellative commutative monoids. In case the monoid is an affine semigroup, we give a geometrical characterization of such submonoids in terms of its cone. Finally, we use our results to give an algorithm for computing $\Delta^*(H)$ the set of minimal distance of $H$.