Unary operations with long pre-periods

Abstract

It is well known that the congruence lattice Con A of an algebra A is uniquely determined by the unary polynomial operations of A (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 [2]]). Let A be a finite algebra with | A | = n . If Im f = A or | Im f | = 1 for every unary polynomial operation f of A , then A 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 [3]]. If f : A → A is not a permutation then A ⊃ Im f and there is a least natural number λ ( f ) with Im f λ ( f ) = Im f λ ( f ) + 1 . We consider unary operations with λ ( f ) = n - 1 for n ⩾ 2 and λ ( f ) = n - 2 for n ⩾ 3 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.