A Moufang loop with exceptional properties of associators

TL;DR

This paper constructs a large nonassociative Moufang loop demonstrating unique associator properties, including a set of elements that do not form a subloop despite associating with specific generators.

Contribution

It presents the first known example of a Moufang loop with a set of elements that do not form a subloop, despite all triples of generators associating.

Findings

Constructed a Moufang loop of order 3^19 with exceptional associator properties

Provided an example where the associator set does not form a subloop

Showed that all triples of generators in the loop associate

Abstract

We construct a Moufang loop $M$ of order $3^{19}$ and a pair $a,b$ of its elements such that the set of all elements of $M$ that associate with $a$ and $b$ does not form a subloop. This is also an example of a nonassociative Moufang loop with a generating set whose every three elements associate.