Let x,y be in M. Then
x*y = x*(y*y) (using (i))= (x*y)*y (associativity)= (y*y)*x (using (ii))= y*x (using (i).
I have the following problem:
Let (M; *) be a groupoid in which the following relations hold:
i) x * x = x; x belongs to M
ii) (x * y) * z = (y * z) * x; x, y, z belong to M.
Show that the operation * is commutative.
I don't know how to start to prove this. Please help Much appreciated!