Universal $\alpha$-elasticity of generalized Moufang loops

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.