Cheban loops

Abstract

Left Cheban loops are loops that satisfy the identity $x(xy \cdot z) = yx \cdot xz$. Right Cheban loops satisfy the mirror identity $(z \cdot yx)x = zx \cdot xy$. Loops that are both left and right Cheban are called Cheban loops. Cheban loops can also be characterized as those loops that satisfy the identity $x(xy \cdot z) = (y \cdot zx)x$. These loops were introduced by A. M. Cheban. Here we initiate a study of their structural properties. Left Cheban loops are left conjugacy closed. Cheban loops are weak inverse property, power associative, conjugacy closed loops; they are centrally nilpotent of class at most two.