Tilings和相关的关系结构
Tilings and associated relational structures
摘要：
在本文中，正如我们以前在[7]中所做的那样，我们研究了镶嵌的几何性质和相关关系结构的代数性质之间的关系。我们的研究是由非周期平铺系统的存在。在[7]中，我们考虑了有限维欧几里得空间的镶嵌，并定义了直到平移的同构。在这里，我们考虑，更一般地说，一个度量空间的平铺，和同构定义模任意组的等距。在第1节中，我们定义了与平铺相关的关系结构。第二节的结果涉及局部同构，提取预序和关系结构的特征，可以表示为一些给定类型的平铺。在第3节中，我们证明了，在适当的假设下，为有限维欧几里得空间的平铺定义的通过平移的周期性和不变性的概念可以推广到关系结构，特别是非欧几里得空间的平铺。第4节的结果对一致局部有限关系结构，特别是满足局部同构性质的平铺结构是有效的。我们的特点是这些结构之间的局部同构的结构没有非平凡的自同构。我们表明，在有限维的欧几里得空间，这个属性是真实的瓷砖满足局部同构性质的当且仅当它不是通过非平凡的平移不变。我们说明这些结果的例子，其中一些关于非周期平铺系统的欧几里德或noneuclidean空间。
英文摘要：
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of aperiodic tiling systems. In [7], we considered tilings of the euclidean spaces of finite dimension, 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. In Section 1, we define the relational structures associated to tilings. The results of Section 2 concern local isomorphism, the extraction preorder and the characterization of relational structures which can be represented by tilings of some given type. In Section 3, we show that the notions of periodicity and invariance through a translation, defined for tilings of the euclidean spaces of finite dimension, can be generalized, with appropriate hypotheses, to relational structures, and in particular to tilings of noneuclidean spaces. The results of Section 4 are valid for uniformly locally finite relational structures, and in particular tilings, which satisfy the local isomorphism property. We characterize among such structures those which are locally isomorphic to structures without nontrivial automorphism. We show that, in an euclidean space of finite dimension, this property is true for a tiling which satisfies the local isomorphism property if and only if it is not invariant through a nontrivial translation. We illustrate these results with examples, some of them concerning aperiodic tilings systems of euclidean or noneuclidean spaces.
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