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 \( \mathbb{T}_{max} \))

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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 \( K \)

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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

http://math.eretrandre.org/tetrationforu...+operators)