the distributive property
a [r] b = b [r] a = exp^[r]( exp^[-r](a) + exp^[-r](b) )

now the extended distributive property is

a [r] ( b [r-1] c ) = ( a [r] b ) [r-1] ( a [r] c )

and once again tetration plays an underestimated important role Smile

it gets really intresting when we set a = b = x and then take the superfunction.

x + x = 2x => 2^r x

x * x = x^2 => x^(2^r)

x ^ ln(x) => x^(ln(x)^(2^r - 1))

and notice the similarity between the last two !!!

( the first two are similar too but less intresting and better known )


perhaps it wasnt clear from the OP but i was playing with the idea of " uniqueness condition " for tetration regarding this " formula "

More precisely :

Uniqueness or at least a strong condition :

Let B2 > B1 > e^(1/e).

Base independance :

r > 0

Define for base C :

exp_C(z) = c^z

a [r]_C b = b [r]_C a = exp_C^[r]( exp_C^[-r](a) + exp_C^[-r](b) )

Conjecture :

a [r]_B2 b = b [r]_B2 a = exp_B2^[r]( exp_B2^[-r](a) + exp_B2^[-r](b) ) = a [r]_B1 b = b [r]_B1 a = exp_B1^[r]( exp_B1^[-r](a) + exp_B1^[-r](b) )

Or simply : a [r]_B2 b = a [r]_B1 b.

Remark : its known to be true for r a positive integer. (it fails for r = -1 though )

Unfortunately I lost some data concerning this idea.

So if you see it mentioned here before , plz let me know.

I am also intrested in other " base independance " equations.


As you know these are a generalization of the Bennet Hyperoperations (Commutative Hyperoperations).

I usually use this notation for them because i find it very confortable

\( a \odot_r^k b:={\exp}_k^{\circ r}(log_k^{\circ r}(a)+log_k^{\circ r}(b)) \)

Bennet Hyperoperations are a special case of these (with the natural base \( a \odot_r^e b \))
\( \odot_0^{K} =+ \)
\( \odot_1^{K} =\cdot \)
and \( a \odot_{-1}^{+\infty} b \) is the max operation while \( a \odot_{-1}^{0} b \) is the min operation(this limit process is related with the litinov-maslov dequantization of the semifield of non-negative real numbers in to the Tropical semifield \( \mathbb{T}_{max} \))

I apologize in advance if I did not understand your conjecture but i think that is not true at all.

In general we have that the operations are different when base changes

\( a \odot_r^{K} b=a \odot_r^{J} b \) only if \( r=0,1 \)

If you quickly plot the graps of \( a \odot_2^{K} x \) for differents bases \( K \) you see that the the result changes.

Notice that the identity element of \( \odot_2^{K} \) is \( K \)

About tetration: yes! There is an interesting link because an extension of tetration will bring to us the fractional rank operations of this Hyperoperation family (JmsNxn already worked on something similar in his thread on logarithmic semi-operators

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
Right sorry , I meant for 0 < r < 1.

I havent given it any time yet , but what happens if the base goes to 1 from above ? In a limit way ofcourse.


Interesting thing. for integers ranks \( r=0 \) and \( r=1 \) the limit
\( lim_{K\rightarrow 1^+}\odot_{r}^K \)
is still addition and multiplication, but between I don't know (but I doubt... I'd except some dequantization phenomenon like for the tropical operations max and min)...
If someone has an extension for tetration to the reals he could check if for that extension the operations \( \odot_{r \in ]0,1[}^K \) satisfie you conjecture.

Probably JmsNxn knows more about it.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
Finally , it all comes down to solving :

f ( f^[-1](a) + f^[-1](b) ) = a + b.

I can explain that if you want, but I think most here understand it by just seeing it.


In fact I don't get it at all, probably is related with iteration theory or idk..

How can help us finding such automorphism of addition? Anyways I guess that probably there are infinite I do not know where you would end up

for example \( f=id \) is a trivial solution.

imho we should just see if

\( ^{(slog_J(a)+0,5)}J + ^{(slog_J(b)+0,5)}J = { ^{(slog_K(a)+0,5)}K + ^{(slog_K(b)+0,5)}K} \)

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
Well we get exp(x) replaced by exp(f(x)) and ln(x) replaced by f^[-1](ln(x)).

Hence we arrive at f ( f^[-1](a) + f^[-1](b) ) = a + b

( with f at least C^3 )

now differentiate with respect to a :

f ' (f^[-1](a) + f^[-1](b)) / f ' (f^[-1](a)) = 1

substitute X = f(a) , Y = f^[-1](b) :

f ' ( a + Y) = f ' (a)^2

Now rewrite f ' = G and differentiate with respect to Y :

G ' (Y + a) = 0

Hence G is constant and thus f = lineair.

Since f(0) = 0 we conclude f(x) = C x.

thus it only holds for r = 0,1




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