Deformations of toric singularities

The possibilities of splitting a lattice polytope with  primitive edges into a Minkowski sum of lattice polytopes reflect the deformation theory of the induced toric Gorenstein singularity. This was proved by Altmann in 1996. Our goal is to reveal a similar correspondence for arbitrary toric singularities. To this aim we prove the existence of an initial object  in the category of extensions of semigroups, and define a lattice structure inside the cone of Minkowski summands.