Exponentiation is distributive over multiplication, but it isn't commutative or associative like addition and multiplication are.

Is there a binary operation that is distributive over multiplication, and also commutative and/or associative?

In order to find one such operation, I assumed that there is an identity element. An easier question than the one above is: Is the identity element 0, 1, or neither?

If you find an operation that works, can you then find a commutative and/or associative operation that is distributive over that?

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