Some fundamental properties of tiling semigroups (Q1849098)

A tile is a connected bounded closed subset of a Euclidean space. A tiling \(\mathcal T\) is an infinite set of tiles covering \(\mathbb{R}^n\), overlapping (at most) at their boundaries; a pattern is a finite subset of a tiling; only connected patterns are considered. On the set \(C({\mathcal T})=\{(p_2,P,p_1)\mid P\) a pattern, \(p_1,p_2\in P\}\) a partial multiplication is defined by \((p_2,P,p_1)(q_2,Q,q_1)=(p_2,P\cup Q,q_1)\), iff \(p_1=q_2\); \(S({\mathcal T})=(C({\mathcal T})/G)^0\), where \(G\) is the equivalence on \(C({\mathcal T})\) defined by the group of all translations of \(\mathbb{R}^n\). Green relations and some congruences on the inverse semigroup \(S({\mathcal T})\) are described.