  Given a Noetherian ring $A$, the collection of all integrally closed ideals
in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative
monoid under the operation $I*J=\overline{IJ}$, the integral closure of the
product. The monoid is torsion-free and atomic -- every integrally closed ideal
in $A$ containing a nonzerodivisor can be factored in this $*$-product into
$*$-irreducible integrally closed ideals. Restricting to the case where $A$ is
a polynomial ring and the ideals in question are monomial, we show that there
is a surjective homomorphism from the Integral Polytope Group onto the
Grothendieck group of integrally closed monomial ideals under translation
invariance of their Newton Polyhedra. Notably, the Integral Polytope Group, the
Grothendieck group of polytopes with integer vertices under Minkowski addition
and translation invariance, has an explicit basis, allowing for explicit
factoring in the monoid.

Commutative Algebra