Unary operations with long pre-periods Original Research Article

It is well known that the congruence lattice image of an algebra image is uniquely determined by the unary polynomial operations of image (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 ]). Let image be a finite algebra with image. If image or image for every unary polynomial operation f of image, then image is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 ]. If image is not a permutation then image and there is a least natural number image with image. We consider unary operations with image for image and image for image and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.