bo198214 Wrote:[...]Hmm, I wasn't aware of this property being referred to as the "distributive law"... I am more familiar with distributive laws being of the form \( a\diamond(b\circ c) = (a\diamond b)\circ(a\diamond c) \) (and/or vice versa for non-commutative operators). But anyway, it's good to clarify what you meant by "distributive law".
I does not so much matter whether you use the logarithm or the exponential itself, what makes exponentiation and multiplication unique are the "distributive laws":
\( a^{x+y}=a^xa^y \) and \( a(x+y)=ax+ay \) which are an extension of
\( a^{x+1}=aa^x \) and \( a(x+1)=ax+a \), however we know it doesnt work for tetration.
Quote:Thatswhy I decided to develop a new more general number domain, arborescent numbers, in which we have this distributivity a[[n+1]](x+y)=(a[[n+1]]x)[[n]](a[[n+1]]y) for arbitrary high operations, as we just discussed here.Right. I'm not entirely convinced, though, that it's not possible to find some kind of "natural" uniqueness property that must hold for tetration, which would uniquely determine it on the reals. It may require operations (or rather, binary functions) other than addition, multiplication, or exponentiation to be expressed, but surely there must be some such property that would allow us to have uniqueness.
Quote:However it would be good if you have a read through the Tetration FAQ, as uniqueness of exponentiation and multiplication are already explained there.Thanks, I just looked over it quickly. I think the uniqueness of exponentiation via what you call the "distributive laws" is quite a well-known property of exponentiation, and indeed, it's probably the reason exponentiation was so well studied in the past (before the advent of computers, converting multiplications to additions via logarithms was a very handy method of speeding up computations). It directly results from multiplication being associative.
I sorta referred to this in my thread on higher-order operations on ordinals... multiplication is the last associative operator in the Grzegorczyk hierarchy, so exponentiation (i.e., a multiplication chain) is the last operation where you can "append to the chain" at either end easily. I.e., since x*x*x*...*x can be bracketed in any order (multiplication is associative), we can easily add another instance of x to the end of the chain by computing x*x*...*x (n times), and then multiplying the result by x. This makes it easy to generalize exponentiation to ordinals: to define exponentiation by \( \omega+1 \), you just compute \( \alpha^{\omega} \), and then right-multiply the result by \( \alpha \).
But for operators with n=4 and above (tetration and above), the previous operators are not associative, so this trick cannot work: there is no algebraic way to "reach the top of the tower" in the expression x^x^x^...^x (assuming right-to-left bracketing). So it's non-trivial to define what is meant by tetrating to \( \omega+1 \). Non-associativity also causes the negation of the so-called "distributive laws", which are essentially a consequence of the associativity of multiplication in the case of exponentiation. There is no algebraic way for us to "reach into" the top exponent of the tower x^x^...^x and add more to it, even though we can right-multiply the tower. If we could reach into the top exponent somehow, then we could have a relation between adding to the tower from the bottom, and adding to it from the top; then that may give us some kind of property that may help determine uniqueness for tetration over the reals. But as far as we know, there is no way to do this, and so no straightforward property we can exploit to determine uniqueness.