Persistence of cyclic left distributive algebras

Let A(k) = A(k)(*) denote the left distributive groupoid on (0,1,...,2(k) - 1} such that a* 1 = + 1 mod 2(k) for every a is an element of A(k). Let d greater than or equal to 0 and put r = max{i;2(i) divides d). For a =Sigma a(i)2(i) is an element of A(k), a(i) is an element of{0,1}, Put nu(d)(a) =Sigma a(i) nu(d)(2(i)) and nu(d)(2(i)) = 2((i+1)2d) -2(i2)d. Then nu(d): A(k) -->A(k2)(d) is a groupoid homomorphism iff k less than or equal to 2(r+1).