Categories of hypermagmas, hypergroups, and related hyperstructures
Abstract
In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas -- which we call mosaics -- form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory.
Cite
@article{arxiv.2304.09273,
title = {Categories of hypermagmas, hypergroups, and related hyperstructures},
author = {So Nakamura and Manuel L. Reyes},
journal= {arXiv preprint arXiv:2304.09273},
year = {2025}
}
Comments
54 pages, 3 figures. Corrections made throughout, but especially to Theorem 1.1, Lemma 2.13, and Theorem 4.19. Added Lemma 4.22 and Proposition 4.23. Final version