Defining a Binary Operation

Algebra Level 5

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

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

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

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


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