I want "something" ("something" because maybe it is not really a mathematical function, called F in the above image) that can describe what is shown on the image. A given value from a domain Xi can be mapped to many values of domain X{i+1}, and for two values $x \neq x'$, then $F(x) \neq F(x')$
It also should be invertible, i.e. $F^{-1}$ exists.
Is it possible to have this ? If not, how can I have an alternative for this with something possible in the mathematics ? Maybe using a function where the elements of the codomain are vectors ? ...
EDIT: I'm not just seeking a name for that. Let's call it a "relation". I want to define a relation F that allow me to generate a set of numbers $y1, y2, y3$ given a number x (that is, F(x) = {y1, y2, y3}), and I want it to be invertible, that is $F^{-1}(y1) = x$, $F^{-1}(y2) = x$ and $F^{-1}(y3) = x$.
This numbers (elements of each domain $X_i$) may be natural, real, complex, or whatever ...
Also, note that the domains $X_i$ does not overlap, that is elements are unique: for any given element x from $X_i$, there is no element y in any other $X_j$ such that x=y.