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Categories of hypermagmas, hypergroups, and related hyperstructures

So Nakamura, Manuel L. Reyes

TL;DR

The paper introduces mosaics, a broad generalization of canonical hypergroups in which the hyperoperation may yield the empty set, and proves that key categorical properties (completeness, cocompleteness, free objects, and closed monoidal structures) hold for hypermagmas, unital hypermagmas, mosaics, and commutative mosaics. It shows canonical hypergroups and hypergroups lack several of these favorable features, justifying mosaics as a more robust framework. A closed monoidal structure on commutative mosaics is developed, with a tensor-like product $\boxtimes$ and an internal hom enabling a tensor-hom-like theory, and a representability result for bimorphisms strengthens this view. The work also builds bridges to matroid theory and projective geometries by embedding simple pointed matroids and projective geometries into mosaic categories, suggesting a versatile, base-free approach to studying hyperrings and multirings via representation theory and categorical methods. Overall, mosaics emerge as a flexible, well-behaved categorical setting for generalized hyperstructures with promising connections to combinatorial geometries and potential applications to hyperrings and representations.

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.

Categories of hypermagmas, hypergroups, and related hyperstructures

TL;DR

The paper introduces mosaics, a broad generalization of canonical hypergroups in which the hyperoperation may yield the empty set, and proves that key categorical properties (completeness, cocompleteness, free objects, and closed monoidal structures) hold for hypermagmas, unital hypermagmas, mosaics, and commutative mosaics. It shows canonical hypergroups and hypergroups lack several of these favorable features, justifying mosaics as a more robust framework. A closed monoidal structure on commutative mosaics is developed, with a tensor-like product and an internal hom enabling a tensor-hom-like theory, and a representability result for bimorphisms strengthens this view. The work also builds bridges to matroid theory and projective geometries by embedding simple pointed matroids and projective geometries into mosaic categories, suggesting a versatile, base-free approach to studying hyperrings and multirings via representation theory and categorical methods. Overall, mosaics emerge as a flexible, well-behaved categorical setting for generalized hyperstructures with promising connections to combinatorial geometries and potential applications to hyperrings and representations.

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.
Paper Structure (12 sections, 48 theorems, 214 equations, 3 figures)

This paper contains 12 sections, 48 theorems, 214 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A brief overview of hyperstructures
  3. Substructures of hyperstructures and associated morphisms
  4. Some examples of hypergroups and hypermagmas
  5. Categories of hypermagmas
  6. Forgetful functors and (co)limits
  7. Characterizations of various morphisms
  8. Closed monoidal structures
  9. Categories of hypergroups and mosaics
  10. A convenient category of (canonical) hypergroups
  11. The category of (canonical) hypergroups
  12. Matroids as mosaics

Key Result

Theorem 1.1

The categories $\mathop{\mathrm{\mathsf{HMag}}}\nolimits$, $\mathop{\mathrm{\mathsf{uHMag}}}\nolimits$, $\mathop{\mathrm{\mathsf{Msc}}}\nolimits$, and $\mathop{\mathrm{\mathsf{cMsc}}}\nolimits$ are complete and cocomplete, and free objects exist in these categories. The categories $\mathop{\mathrm{\

Figures (3)

  • Figure 1: Categories of hyperstructures. A "$\subseteq$" denotes a full subcategory, while the "$\hookrightarrow$" is a faithful forgetful functor.
  • Figure 2: Characterizations of various monomorphisms.
  • Figure 3: Characterizations of various epimorphisms.

Theorems & Definitions (115)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 1.4
  • proof
  • Theorem 1.5: Theorem \ref{['thm:multirings']}
  • Theorem 1.6: Theorems \ref{['thm:matroid mosaics']} and \ref{['thm:projective embedding']}
  • Definition 2.2
  • Definition 2.3
  • ...and 105 more