On Groupoids and Hypergraphs

TL;DR

This paper introduces a new method for constructing finite groupoids with large girth in their Cayley graphs, enabling the creation of highly symmetric hypergraphs with specific overlap patterns and no small cycles.

Contribution

It presents a novel construction of finite groupoids with large girth and applies this to generate hypergraphs that realize specified overlaps while avoiding short cycles.

Findings

Constructed finite groupoids with large girth under a discounted distance measure.

Developed a generic method for hypergraph construction with prescribed overlap patterns.

Produced highly symmetric hypergraph coverings without small cycles.

Abstract

We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified overlap patterns and avoid small cyclic configurations. The constructions are based on reduced products with groupoids generated by the elementary local extension steps, and can be made to preserve the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing operations and require global finite closure.