On left alternative loops

A groupoid ) , ( Q is a quasigroup if, for each Q b a , , the equations b ya b ax , have unique solutions where Q y x , [1]. A loop is a quasigroup with an identity element e such that x e x e x . The left nucleus of a loop Q is } , ) ( ) ( : { Q y x y lx xy l Q l N . The right nucleus of a loop Q is the set } , ) ( ) ( : { Q y x yr x r xy Q r N , and middle nucleus of Q is } , ) ( ) ( : { Q y x mx y x ym Q m N . The nucleus of Q is the set N N N N [2, 3]. A loop ) , ( L is termed as left alternative loop if the following identity is satisfied for all L z y x , , :