Commutative Moufang loops and alternative algebras

TL;DR

This paper investigates the structure of free commutative Moufang loops of exponent 3, computing their orders for up to 7 generators and exploring their embeddings into free commutative alternative algebras.

Contribution

It provides explicit order computations for these loops and demonstrates their embeddings into invertible elements of free commutative alternative algebras.

Findings

Computed orders of free commutative Moufang loops for n ≤ 7 generators.

Established embeddings into invertible elements of free commutative alternative algebras.

Enhanced understanding of the algebraic structure of these loops.

Abstract

We compute the orders of free commutative Moufand loops of exponent 3 with $n\leq 7$ free generators and find embeddings of such loops into a loop of invertible elements of the free commutative alternative algebra with identity $x^3=0$.