Role of xor in ARX construction In ARX construction like Salsa 20, why xor operation is required? That is why AR is not sufficient? Note that xor is a linear operation.
Xor is the "addition" operator for algebra with boolean operators, while AND is the "multiplication" operator for boolean operators. "Regular" addition and multiplication are the addition/multiplication operators for integers. So speaking of "linearity" in the sense of algebraic compatibility, XOR and integer addition are non-linear. I feel like this question/answer(s) sums it up pretty well. 

As for why xor is required/why AR is not sufficient, technically, AR is equivalent to ARX, but less efficient. See the paper Rotational Cryptanalysis of ARX . There are some details to it:


  We also show that the
  AR systems
  , that do not use XOR, are theoretically
  equivalent to ARX systems. However, we prove that they are less secure with
  the same number of operations, because of the linear mod 2**n approximation.
  It is also easy to prove that omitting addition or rotation is devastating, and
  such systems (XR and AX) can always be broken.