Polyhedra, lattice structures, and extensions of semigroups

For an arbitrary rational polyhedron, we consider its decompositions into Minkowski summands and, dual to this, the so-called free extensions of the associated pair of semigroups. Being free for a pair of semigroups is equivalent to flatness for the corresponding algebras. The main result is phrased in this dual setup: the category of free extensions always contains an initial object, which we describe explicitly. This provides a canonical free extension of the original pair of semigroups provided by the given polyhedron. Our motivation comes from the deformation theory of the associated toric singularity.