A More Symmetric Exponentiation

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?

• a^(bc) = (a^b)(a^c) is what "exponentiation is distributive over multiplication" ought to mean, if it is supposed to be like multiplication over addition. But .... – Ned Mar 1 '18 at 21:53

For positive real numbers, you can define

$$a * b = e^{\log(a) \cdot \log(b)}.$$

This is a commutative and associative operation, it has an identity element $e$, and it distributes over multiplication:

$$a * (b\cdot c) = (a * b) \cdot (a * c),$$

as is easily checked. In fact there is an infinite family of such operations, as you can replace $e$ by any positive constant $k\neq 1$, and $\log$ by $\log_k$.

For a commutative and associative operation that distributes over $*$, you can define

$$a ** \:b = e^{\log(a) * \log(b)} = e^{e^{\log(\log(a)) \cdot \log(\log(b))}},$$

which has an identity $e^e$. It is clear that you can keep building new operations this way indefinitely, forming a commutative version of the hyperoperation sequence.