Zeration

[marraco]

f=f+0
f°f=f+1
(f°f) ° (f°f)=f+2
((f°f) ° (f°f)) ° ((f°f) ° (f°f)) =f+3
(f+n)°(f+n)=f+(n+1)

and the natural choice for f, should be the neutral element of addition, so
0 = 0
0°0 = 1
(0°0) ° (0°0) = 1°1 = 2
((0°0) ° (0°0)) ° ((0°0) ° (0°0)) = 2°2 =3
(0+n)°(0+n) = n°n = (n+1)

let's call \( \circ{n} \,=\, (\cdots (((-\infty \,\circ \, 0_1) \,\circ \, 0_2) \,\circ \, 0_3) \, \cdots \,\circ \, 0_n) \)

then we have:

○0=-∞
○1=0
○2=1
○3=1○0
○4=2
○5=○4○0
○6=○5○0
○7=○6○0
○8=○7○0
○16=4
⁞
○2 ͫ =m (this may only be valid for m ∈ ℤ)

¿can this zeration be associative?

If it is associative, then this is a zeration table for °0 to °16



The blue numbers are the natural numbers. I don't know if in general ○n ∈ ℝ, so I wrote \( \circ{n} \,=\, (\cdots (((-\infty \,\circ \, 0_1) \,\circ \, 0_2) \,\circ \, 0_3) \, \cdots \,\circ \, 0_n) \)

If "our" logarithm is the same that "zeration" logarithm (which is dubious), then
a°b=ln₂(2ᵃ+2ᵇ)
○a°○b=○(a+b)

A question arises: can the symbol ○ ,used as a sign, be identified with a number or an operation?
the symbol "-" can be identified with -1, and "+" with +1 and 0+
-n = -1.n
+n = 0+n
We can also use (/n = 1/n) or (÷n = 1/n)
or (±in = ±i1.n)
or Lm=ln(m)

we can combine and operate those symbols -i = /i = ÷i

We can also define the operation inverse of zeration this way (☺n=InverseOf(n) ⇔ ☺n○n = -∞) or (☺n = -∞☺n)

(Sorry, I didn't found the combination of - and ○)

Does this mean anything: -☺/i n ?
Is (-☺/i n = ☺i/- n) ?

From the table,
○-∞=-∞
○0=-∞
○1 = 0
○2 = 1
○4 = 2
○2ⁿ=n

but this is associated to a capricious choice of f and 2. Maybe is more "natural" to use
(○eⁿ = n) or (○n = ln(n))

That would make
○e = 1 + i2.n.\( ^{\pi} \)
○-1 = i\( ^{\pi} \)(1+2n)
○i = i\( ^{\pi} \)(½+2n)
e°⁻¹+1 = ○1

^ There I'm "ignoring" that "our" logarithm may not be the same thing as "zeration" logarithm.

maybe it should be necessary to specify what f and n numbers are used
\( a \,\circ_f^n\, b \)
where f is the first number known by "zerationists", and n the one of "additionists", from "zerationists" viewpoint.

the natural zeration would be
\( a \,\circ_{\small{0}}^{\small{e}}\, b \)
or
\( a \,\circ_{\small{0}}^{\small{2}}\, b \)

This shows that zeration may be as complex as exponentiation, so maybe addition is the real "zero" operation (the simplest one), and zeration is the "negation", or "real zeration" lies between addition and product.
\( ^{ \circ_{-1} \,\leftarrow\, +_0 \,\rightarrow\, \times _1 \,\rightarrow\, \wedge\,_2 \,\rightarrow\, \wedge\wedge\,_3} \)
or
\( ^{ \circ_{-2} \,\leftarrow\, +_{-1} \,\leftarrow\, \small{''real\,zeration''}_0 \,\rightarrow\, \times _1 \,\rightarrow\, \wedge\,_2 \,\rightarrow\, \wedge\wedge\,_3} \)