Differentiable monoids with smooth boundary

We investigate here the structure of monoids which are based upon differentiable manifolds with smooth boundary and which have differentiable multiplication function. If the multiplication function of such a monoid, S, is C^k for k≥2 and the unit element, 1, is in the boundary then the maximal subgroup, H(1), of S containing 1 is a C^k group which is an open subgroup of the boundary of S.