  We study the universal groups of inverse semigroups associated with point
sets and with tilings. We focus our attention on two classes of examples. The
first class consists of point sets which are obtained by a cut and projection
scheme (so-called model sets). Here we introduce another inverse semigroup
which is given in terms of the defining data of the projection scheme and
related to the model set by the empire congruence. The second class is given by
one-dimensional tilings.
