Zeration

[quickfur]

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Eventually, I came upon the definition: \( a\diamond b = \ln(e^a + e^b) \). As you can verify, \( a+(b\diamond c) = \ln\ e^a + \ln(e^b + e^c) = \ln(e^a(e^b + e^c)) = \ln(e^{a+b}+e^{a+c})=(a+b)\diamond(a+c) \).

Yes, you are right, that it is possible to construct this operator. It is a classical homomorphism of addition with using function exp(x). I researched in 90th years application of this operator and of some others for upgrade numerical solution systems of the differential equations. Good the outcome was with a homomorphism on the basis function ln(x).
However, this operator (a homomorphism of addition) is not Zeration. Moreover, application Zeration has allowed to achieve more simple algorithms of a homomorphism methods numerical solution of the differential equations and their systems. These researches are available in my monography (in Russian).

Я согласен с Вами. My operation is not zeration, even if it does exhibit some similar properties, such has having identity element as \( -\infty \). As I noted, my operator does not satisfy the defining property of the operations hierarchy, namely, that the n'th iteration of it should correspond to addition by n. Zeration as you have defined does have this property, which is good.

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If Zeration it is commutative \( a\Delta a = \left( { - \infty } \right)\;\; \Rightarrow \;\left( { - \infty } \right) \circ a = a \).

Yes, it seems that any operation below addition, where some form of iteration reduces to addition, must have \( -\infty \) as its identity element. As I've noted, any operation # which qualifies as being "before addition" must have as identity something that behaves like a multiplicative zero with respect to addition. \( -\infty \) qualifies, because, at least intuitively speaking, \( (-\infty)+x = -\infty \).

The interesting thing about this, is that if we then construct inverse elements w.r.t. to #, then we must admit new numbers that lie "before" \( -\infty \). This seems quite reminiscient of how constructing the inverse of addition created the negative numbers, the inverse of multiplication created the rational numbers, and the (radical) inverse of exponentiation created the real numbers (due to such constructs as \( \sqrt{2} \)). Combining the (radical) inverse of exponentiation with the negative numbers gave us the complex numbers. One can hardly wonder that the inverse of zeration would yield new numbers too. (It makes one wonder if the inverse of tetration would also create new numbers... I suspect it must've come up in this forum before, right?)

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P.s. I bring an apology, for quality of computer translation on English.

Нет ничего. As long as it gets the message across... (Now it's my turn to apologize for my limited Russian.)