Steiner Loops Satisfying the Statement of Moufang's Theorem

TL;DR

This paper investigates whether certain Steiner loops satisfy Moufang's theorem and provides examples of Steiner loops that do and do not satisfy the theorem, addressing a conjecture about their relationship to Moufang loops.

Contribution

It offers a partial answer to a conjecture by presenting specific Steiner loops that either satisfy or do not satisfy Moufang's theorem.

Findings

Identified Steiner loops satisfying Moufang's theorem

Constructed Steiner loops not satisfying Moufang's theorem

Showed these loops are not Moufang loops

Abstract

Andrew Rajah posed at the Loops'11 Conference in Trest, Czech Republic, the following conjecture: Is every variety of loops that satisfies Moufang's theorem contained in the variety of Moufang loops? This paper is motivated by that problem. We give a partial answer to this question and present two types of Steiner loops, one that satisfies Moufang's theorem and another that does not, and neither is Moufang loop.