In previous time I supposed to make a paper in English explaining my thoughts about zeration, but now I nly am able to present here an article in Russian. It is not provided with a literature overview but I've read some works by G.F. Romerio & Rubtsov, and some other works named in my paper. Finally I have to note that there's a lot of questionable points in my suggestions about zeration and I am ready to discuss them.

UPD. Some questionable points.

1) Our resulting solution (an algorithm for calculating collection operation) is depending on what requirements we want to satisfy. E. g., I wanted to achieve an implicit algorithm based on hyperboloidous approximations, plus some 'common sense' suppositions - so then I've got my algorithm which is in this paper. But nevertheless I've also got another, explicit formula with exponents, satisfying all the constarints too, but in another manner. Furthermore, I can put some new constraints built on a sand, in a manner 'what if also ...', in a 'common sense' manner too, which are making collection operation more similar to summ and multiplication but are not allowing to use my algorithm from the paper.

So the question is what do we want? What is the only set of requirements necessary for an operation, such that we could call it pre-addition?

My first algorithm presented in the paper is based on polynoimal equations and hence leads to everywhere we want in mathematics.

My second algorithm not presented here is based on exponential functions and allows to connect this operation to complex numbers, Fourier transformation and so on.

A way chosen by Romerio & Rubtsov, max(a, b) + 1, tends to give non-smooth functions and thus is harder to extend I guess.

Is it probable to connect these and some other ways together in order to find a non-controversary image of a pre-addition?

We all would like to finally get an infinite array of numerical operations or, probably, an infinite tree or multidimensional set of them, but we should list all the requirements we want to satisfy, and issue all the features we are able to discover in the operations which are already known to us.

UPD. Some questionable points.

1) Our resulting solution (an algorithm for calculating collection operation) is depending on what requirements we want to satisfy. E. g., I wanted to achieve an implicit algorithm based on hyperboloidous approximations, plus some 'common sense' suppositions - so then I've got my algorithm which is in this paper. But nevertheless I've also got another, explicit formula with exponents, satisfying all the constarints too, but in another manner. Furthermore, I can put some new constraints built on a sand, in a manner 'what if also ...', in a 'common sense' manner too, which are making collection operation more similar to summ and multiplication but are not allowing to use my algorithm from the paper.

So the question is what do we want? What is the only set of requirements necessary for an operation, such that we could call it pre-addition?

My first algorithm presented in the paper is based on polynoimal equations and hence leads to everywhere we want in mathematics.

My second algorithm not presented here is based on exponential functions and allows to connect this operation to complex numbers, Fourier transformation and so on.

A way chosen by Romerio & Rubtsov, max(a, b) + 1, tends to give non-smooth functions and thus is harder to extend I guess.

Is it probable to connect these and some other ways together in order to find a non-controversary image of a pre-addition?

We all would like to finally get an infinite array of numerical operations or, probably, an infinite tree or multidimensional set of them, but we should list all the requirements we want to satisfy, and issue all the features we are able to discover in the operations which are already known to us.