2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129897Here we prove that for dilatation structures linearity (see arXiv:0705.1440v1) is equivalent to a statement about the inverse semigroup generated by the family of dilatations of the space. The result is new for Carnot groups and the proof seems to be new even for vector spaces.larger versionGroup TheoryMetric Geometry20M18; 22E20; 20F65text