On the Number of Conjugates of N-Ary Ouasigroups

Abstract

Higher dimensional quasigroups (a set Q with a cancellative, n-ary operation 〈 〉, ([2]) have been studied by T. Evans ([3], [4]), A. Cruse [1], C. C. Lindner ([10], [11]) and also by many others under the guise of magic cubes, Graeco-latin cubes, etc. Conjugates or parastrophes have been discussed by S. K. Stein [18], A. Sade [17] and more recently by C. C. Lindner and D. Steedley in [14], where it is shown that ordinary quasigroups exist of every order ≧ 4 with a prescribed number of distinct conjugates. It is suggested that the problem be extended to n-ary quasigroups.