Strongly differentiable monoids with smooth boundary (Q912990)

The author gives an affirmative answer to a question proposed by \textit{J. P. Holmes} [Semigroup Forum 36, 211-222 (1987; Zbl 0627.22002)] related to the structure of differentiable monoids which are based upon differentiable manifolds with smooth boundary. The main result shows that if the multiplication function \(V\) of such monoid \(D\) is strongly differentiable at 1 (the identity element) and 1 is contained in the boundary of \(D\) then some neighborhood of 1 in \(H(1)\) (the maximal subgroup of \(D\)) is a neighborhood of 1 in the boundary of \(D\).