Tilings and Associated Relational Structures

In the present paper, as we did previously in [5], we investigate the relations between the geometric properties of tilings and the algebraic and model-theoretic properties of associated relational structures. Our study is motivated by the existence of aperiodic tilings. In [5], we considered tilings of the euclidean spaces R, and isomorphism was defined up to translation. Here, we consider, more generally, tilings of a metric space, and isomorphism is defined modulo an arbitrary group of isometries. The results of Sections 1 and 2 concern, in particular, the characterization of relational structures which can be represented by tilings of some given type, local isomorphism and the extraction preorder. In Section 3, we show that the notions of periodicity and invariance through a translation, defined for tilings of the euclidean spaces R, can be generalized, with appropriate hypotheses, to relational structures, and in particular to tilings of non-euclidean spaces. In Section 4, we consider connected relational structures which are locally isomorphic to structures without non-trivial automorphism. We obtain a complete characterization in the case of uniformly locally finite structures which satisfy the local isomorphism property. We give several examples, some of them obtained by considering relational structures associated to tilings of euclidean or non-euclidean spaces. MSC: 52C23 (05B45, 52C22)