Groupoids, Hypergraphs, and Symmetries in Finite Models

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. The universal algebraic and combinatorial properties of groupoids are abstracted form the composition behaviour of partial injections and support a very natural approach to the construction of certain highly symmetric finite instances of hypergraphs and relational structures. The typical task of this kind asks for regular realisations of a locally specified overlap pattern between pieces (hyperedges, guarded substructures). We show that reduced products with groupoids provide a generic and versatile tool towards such constructions; they are explored in applications to the construction of finite hypergraph coverings, to finite model constructions for the guarded fragment, and to extension properties for partial isomorphisms of relational structures (in the sense of Hrushovski, Herwig, Lascar). 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 edges from the same sub-groupoid (colour) and only counts transitions between cosets (different colours), and show that their acyclicity properties guarantee corresponding degrees of acyclicity in reduced products.