The central objects of this talk are pairs of semigroups (T, S), where T is embedded as a subsemigroup of S. A pair is called free if the algebra corresponding to S is flat over the algebra corresponding to T. The main result states that free pairs admit a universal extension. We then focus on semigroups arising from toric geometry, namely semigroups defined by rational polyhedra. In this setting, we present a very explicit description of the universal extension and show how free extensions are dual to Minkowski decompositions of the original polyhedron. The motivation for these constructions comes from the deformation theory of toric singularities and the results presented are based on joint work with Klaus Altmann and Matej Filip.