Some Congruences Associated with the Equation X = X in Certain Finite Semigroups

Let H be a finite group, Tn the symmetric semigroup of degree n, and let α, β be integers with 0 < α < β. We establish congruences modulo an arbitrary prime for the number sn(α, β,H) of solutions of the equationX α = X in the wreath product H oTn, where n = p, p+1, p+2, p+3. Our results generalise well-known congruences for the number of idempotent elements in Tn, to which they reduce for α = 1, β = 2, and H the trivial group. Our approach is based on the exponential generating function for the sequence (sn(α, β,H))n≥0, which is computed, among other things, in J. Combin. Theory (A) 105 (2004), 291–334. Bell’s formalism of exponential polynomials is used to reduce the complexity of the necessary mod p calculations. Key-words: equations in symmetric semigroups, wreath products, congruences modulo primes, idempotents, exponential polynomials. MSC-Index: 05A15, 11A07, 20M20