Between the ops

[Xorter]

I suppose that between evertyhing is there something. E. g. xy = x*y (multiplication).
I suppose that between every operators must be there a multiplication, let us call it operational multiplication: □
For example:
xy = x*y
123 = 1*100 + 2*10 + 3
¬x = ¬ □ x
¬¬x = ¬ □ ¬ □ x
(x = y) = (x □ = □ y)
etc ...
And as the "normal" multiplication has power and functional multiplication (f o c = f( c )) has functional power (f^oN = f o f o ... o f) as the operational multiplication has operational power and roots: O □ O □ ... □ O = O^□N
For instance:
(¬¬x)^□0.5 = x or ¬x, because id id x = x and ¬¬x = x, right?
What do you think, is it exist or not? Can we substitute operational multiplication with other multiplication, like the functional or not?

[Gottfried]

Hmm, Xorter, I'm not sure I do actually understand what you're after. But well, in case this is in line...          

I've one time considered the relation between and as interpolatable. The key was that   and the addition can be written as compared with the multiplication . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of and .                     
Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the and -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz.

Gottfried

[Xorter]

Hmm, Xorter, I'm not sure I do actually understand what you're after. But well, in case this is in line...          

I've one time considered the relation between and as interpolatable. The key was that   and the addition can be written as compared with the multiplication . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of and .                     
Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the and -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz.

Gottfried

Gottfried!
I am afraid your topic is not so relevant or close to mine, to this one. But never mind (but please, open a new topic for it). Maybe I was not so clear.
Of course, I am answering for you according to my knowledge.
If I understand your point, then x[.5]y should be equal to log^o.5(exp^o.5(x)+exp^o.5(y)) (between the addition and multiplication), right?
3+3 = 6
3*3 = 9
And look at this function:
https://www.dropbox.com/s/xr0swdwug44abt...5.jpg?dl=0

The fractional (half-)iterate of exp and log functions according to whose graph I created 3[.5]3 is about 4.62197 < 6 which is a paradox if we suppose that x[z]y < x[w]y if z<w, right?
(2exp^o.5(3) = ~11.9136)
I am afraid the formula we would like to use is wrong.
To be honest I thought of the operators between operators, like between two negations...

[Gottfried]

Upp, so I seem to have missed the point completely. Sorry!

Gottfried

[Xorter]

Upp, so I seem to have missed the point completely. Sorry!

Gottfried

No problem, my friend.
This is why I am trying to explain it clearlier.
Well, I suppose that an operator must exist between the others (other ops and numbers). For instance: between two tensors: O O x = O^2 x = O □  O x, but what is this? Is it the same as which is between a number and the '='?: So 2 □ =
This is not so clear like the fractional iterates of hyper ops. But, I think this way is more closeable/easier.