Universal $\alpha$-elasticity of generalized Moufang loops

TL;DR

This paper introduces the concept of $oldsymbol{ extalpha}$-elasticity in generalized Moufang loops, providing conditions for universality and exploring related algebraic properties and special cases.

Contribution

It presents the first comprehensive analysis of $oldsymbol{ extalpha}$-elasticity in generalized Moufang loops, including universal conditions and new $oldsymbol{ extalpha}$-alternative laws.

Findings

Derived necessary and sufficient conditions for $oldsymbol{ extalpha}$-elasticity to be universal.

Identified conditions under which generalized Moufang loops form abelian groups.

Explored properties of generalized Moufang loops using new $oldsymbol{ extalpha}$-alternative laws.

Abstract

In this study we introduce $\alpha$-elasticity property for generalized Moufang loops. Necessary and sufficient conditions for $\alpha$-elasticity of generalized Moufang loops to be universal are given. Using the universal conditions, and in some cases, with the newly introduced right and left $\alpha$-alternative laws for generalized Moufang loops, some properties of generalized Moufang loops are studied. Condition under which the generalized Moufang loop is an abelian group is stated.