Zeration reconsidered using plusation.
#1
X^^1 = x
X^1 = x
X*1 = x
X(+)1 = x

(+) is plusation.

It came to me that zeration ideas failed or were nonintresting because of using simple addition.

Therefore i now consider this plusation which is more logical.

X(+)2 = x + 1

For c real

X(+)(2^c) = x + c.

= x +1 (c times)

Notice x * 2^y = x + x (y times)

X ^ 2^y = x^2 (y times)

So plusation is a Natural logical operator in the list of noncommutative hyperoperators.

Plusation is like the superfunction of zeration.

Regards

Tommy1729
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#2
It is not exactly the supefunction of zeration (max-kroenecker delta definition)...but it is the superfunction of Bennett's base 2 preaddition \( \odot_{-1} \) (-1th rank in base 2 commutative hos hierarchy).

\( A\odot_{i} B=\exp_2^{\circ i}[\log_2^{\circ i}(A) +\log_2^{\circ i}(B)] \)
\( A\odot_0 B=A+B \)
\( A\odot_{-1} B=\log_2(2^A +2^B) \)

In fact you are right: define plusation (set it as rank 1 of a new sequence)

\( b(+)_1x=b+\log_2(x) \)
\( b(-)_1 x=2^{x-b} \)

Let's take it's subfunction in the variable x (i.e.\( F\mapsto f \) where \( F(x+1)=f(F(x)) \) )

\( b(+)_0x=b(+)_1(1+b(-)_1 x)=\log_2(2^{x}+2^b)=b\odot_{-1}x \)


But the sequence \( (+)_t \) doesn't seem much interesting or natural imho... the only nice properties are that
1) it is based on the usual recursion/iteration law (ML \( b*_{i+1}x+1=b*_i(b*_{i+1}x) \) )
2) it intersects the 2-based commutative hos sequence at t=0 (\( b(+)_0x=b\odot_{-1}x \))


MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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