Free Bol loops of exponent two

TL;DR

This paper constructs free Bol loops of exponent two, providing a canonical form for their elements and describing their multiplication law, advancing understanding of algebraic structures satisfying Bol identities.

Contribution

It introduces a method to explicitly construct free Bol loops of exponent two and defines a canonical form for their elements.

Findings

Canonical form for elements of free Bol loops of exponent two

Explicit multiplication law based on the canonical form

Framework for further algebraic exploration of Bol loops

Abstract

A Bol loop is a loop that satisfies the identity $x((yz)y)=((xy)z)y$. In this paper, we give a construction of the free Bol loops of exponent two. We define a canonical form of all their elements and describe their multiplication law based on this form.