Relation To Normal Monoids
Any cancellative commutative monoid M can be embedded into an abelian group. More precisely, the canonical map from M into its Grothendieck group K(M) is an embedding. Define the normalization of M to be the set
where nx here means x added to itself n times. If M is equal to its normalization, then we say that M is a normal monoid. For example, the monoid Nn consisting of n-tuples of natural numbers is a normal monoid, with the Grothendieck group Zn.
For a polytope P ⊆ Rk, lift P into Rk+1 so that it lies in the hyperplane xk+1 = 1, and let C(P) be the set of all linear combinations with nonnegative coefficients of points in (P,1). Then C(P) is a convex cone,
If P is a convex lattice polytope, then it follows from Gordan's lemma that the intersection of C(P) with the lattice Zk+1 is a finitely generated (commutative, cancellative) monoid. One can prove that P is a normal polytope if and only if this monoid is normal.
Read more about this topic: Normal Polytope
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