For a (partial) action of a group \(G\) on a category \(S\) a relation \(\sim\) is defined on \(S\) by: \(x\sim y\) if there exists \(g\in G\) such that \(g\cdot x= y\). Under the conditions considered by the authors \(\sim\) is an equivalence relation, and a suitably defined partial binary operation turns the set of equivalence classes into a semigroup \(S/G\). In this paper, the semigroups obtained in this way are mainly strongly \(F^*\)-inverse semigroups: they are inverse semigroups where each element is below a unique maximal element with respect to the natural partial order. The paper is concerned with classes of such inverse semigroups obtained from tilings and point-sets.
Let \(S\) be a set equipped with a partial binary operation \(\circ\). On the free group on \(S\) consider the congruence relation generated by the set of pairs \((st,s\circ t)\) for which \(s\circ t\) is defined in \((S,\circ)\). The quotient group modulo this congruence is called the universal group of \((S,\circ)\). If \(S\) is a strongly \(F^*\)-inverse semigroup, then a partial binary operation \(\circ\) is defined on the set \(M(S)\) of maximal elements of \(S\) and it is shown that the universal group of \((S,\circ)\) is the same as the universal group of \((M(S),\circ)\).
For a group \(H\), \(G\) a subgroup of \(H\) and \(X\) a subset of \(H\) containing the identity, a category \(C\) is constructed on which \(G\) acts partially. The corresponding strongly \(F^*\)-inverse semigroup \(C/G=\Gamma(X,G,H)\) is constructed, its universal group is investigated, and a condition is given for this universal group to be \(G\). Given a point-set \(\mathcal D\) (i.e. a subset of \(\mathbb{R}^n\)), a category \(C(\mathcal D)\) is constructed and again a strongly \(F^*\)-inverse semigroup \(\Gamma({\mathcal D})=C({\mathcal D})/\mathbb{R}^n\) is obtained. The universal group of \(\Gamma(\mathcal D)\) is investigated using the partial binary algebra consisting of the maximal elements. Explicit computations are made for point-sets in \(\mathbb{R}\).
Let \(H\) be a locally compact Abelian group, \(\Lambda\) a subgroup of \(\mathbb{R}^d\times H\) such that \(\mathbb{R}^d\times H/\Lambda\) is compact, \(\pi\) and \(\pi'\) the projections of \(\mathbb{R}^d\times H\) onto \(\mathbb{R}^d\) and \(H\) respectively, and \(K\) a nonempty bounded subset of \(H\). Then \({\mathcal D}_K= \{\pi(x)\mid x\in\Lambda\), \(\pi'(x)\in K\}\) is a model set. Some further natural conditions are listed for the ingredients of this construction and it is shown that under these circumstances, \(\Gamma(K,\pi'(\Lambda),H)\cong\Gamma({\mathcal D}_K)/\mathcal E\), where \(\mathcal E\) is an idempotent pure congruence. It follows that both semigroups have the same universal group. Conditions are given for this universal group to be isomorphic to \(G\).
The final section deals with the computation of the universal groups of semigroups for one-dimensional tilings.