Defining a Binary Operation

Algebra Level 5

A binary operation \otimes on a set A, is a function which takes inputs a,bAa, b\in A and produces an output abA a \otimes b \in A. For how many positive real values kk does there exist a binary operation \otimes on the set [0,1] [0,1] such that the following properties hold for any x,y,z[0,1] x, y, z \in [0,1] :

(A)\textbf{(A)} x1=1x=x x \otimes 1 = 1 \otimes x = x,

(B)\textbf{(B)} x(yz)=(xy)z x \otimes ( y \otimes z) = (x \otimes y) \otimes z ,

(C)\textbf{(C)} (zx)(zy)=zk(xy) (zx) \otimes (zy) = z^k (x \otimes y) .

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