Finite Groupoids, Finite Coverings and Symmetries in Finite Structures
Abstract
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.
Keywords
Cite
@article{arxiv.1404.4599,
title = {Finite Groupoids, Finite Coverings and Symmetries in Finite Structures},
author = {Martin Otto},
journal= {arXiv preprint arXiv:1404.4599},
year = {2024}
}
Comments
The construction of finite n-acyclic groupoids in Section 2.4 is flawed and I know of no direct repair: completion turns out to be incompatible with restriction to proper subsets of the generator set, so that the induction towards Proposition 2.22 does not stabilise as claimed. This problem has been overcome in arxiv:1806.08664. Also compare arxiv:1709.00031 and arXiv:2208.03273