Associating hypergraphs defined on loops

TL;DR

This paper introduces a new hypergraph structure on loops, specifically on Moufang loops derived from dihedral groups, and explores its properties.

Contribution

It defines the associating hypergraph on loops and investigates its properties on Moufang loops related to dihedral groups.

Findings

Properties of associating hypergraphs on Moufang loops analyzed

Hypergraph structure linked to dihedral group properties

Potential applications in algebraic and combinatorial structures

Abstract

In this paper, we define a new hypergraph $\mathcal{H(V,E)}$ on a loop $L$, where $\mathcal{V}$ is the set of points of the loop $L$ and $\mathcal{E}$ is the set of hyperedges $e=\{x,y,z\}$ such that $x,y$ and $z$ associate in the order they are written. We call this hypergraph as the associating hypergraph on a loop $L$. We study certain properites of associating hypergraphs on the Moufang loop $M(D_n,2)$, where $D_n$ denotes the dihedral group of order $2n$.