We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual sense, but with respect to a discounted distance measure that contracts arbitrarily long sequences of edges within the same sub-groupoid (coset) and only counts transitions between cosets. Reduced products with such groupoids are sufficiently generic to be applicable to various constructions that are specified in terms of local glueing operations and require global finite closure. We here examine hypergraph coverings and extension tasks that lift local symmetries to global automorphisms.

翻译：我们提出了一种基于群胚的Cayley图约化积来构造有限超图及关系结构的新方法。为此，我们构造了Cayley图具有大围长的群胚——此处的大围长并非通常意义下的概念，而是基于一种折扣距离度量：该度量将同一子群胚（陪集）内任意长的边序列压缩为单次跃迁，仅统计不同陪集间的转移。此类群胚的约化积具有充分的泛化性，可应用于以局部粘合操作表述且需全局有限闭包的各种构造中。本文重点研究了超图覆盖问题及将局部对称性提升为全局自同构的扩展任务。