Modular structures are crystal structures built by subperiodic (zero-, mono- or diperiodic) substructures, called modules. The whole set of partial operations relating substructures in a modular structure build up a groupoid; modular structures composed of identical substructures are described by connected groupoids, or groupoids in the sense of Brandt. A general approach is presented to describe modular structures by Brandt's groupoids and how to obtain the corresponding space groups, in which only the partial operations that have an extension to the whole crystal space appear.

中文翻译：

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模块化结构的类群描述。

模块化结构是由称为模块的亚周期（零，单或二周期）子结构构建的晶体结构。模块化结构中与子结构相关的整个局部操作集合构成了类群。由相同子结构组成的模块化结构由相连的类群或布兰特意义上的类群描述。提出了一种通用方法来描述由Brandt的类群表示的模块化结构以及如何获得相应的空间组，其中仅出现对整个晶体空间具有扩展性的部分操作。