Engel quasigroups associated with cycle systems

Many authors have studied methods of generalising Steiner quasigroups by using certain cycle systems. The usual aims have been to preserve the law (xy)y = x, by using the standard construction, or to preserve the law xy = yx, by using the opposite vertex construction. In this paper we seek to generalise the law (xy)y = x to laws of the form ( ... (xy)y . . . )y = x and (y( ... (y(yx)) ... ) = x (reminiscent of the Engel laws which play such an important role in the study of Burnside groups).