Problem about composition operator

Let S is the set of all R^2 -> R functions.
Let O is S^3 -> S function such that for every f, g, h in S and every
x, y in R:

O(f, g, h)(x, y) = f(g(x, y), h(x, y))

Let T is S -> S function such that for every f, g, h in S:

T(O(f, g, h)) = O(T(f), T(g), T(h))

The identity function on S is a solution. Does there exist other non-
constant solutions?
If so, can you give an example?

If R is replaced with set X such that |X| = 2 then there exists at
least one non-constant solution which is not an identity function. It
is a bijection.

Thank you,
Theron

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Relevant Pages

  • Re: The strange set in R^2
    ... D is then the graph of a bijection f ... bijection between 2 subsets of Q = the rationals, ... Next take the identity function on R-Q, and union that with A the ...
    (sci.math)
  • Re: Cantor Confusion
    ... The countability of the set W of all the words of a finite alphabet is ... impossible to construct a bijection between W and N. Why should it be ... Cantor insisted on well-ordering of all sets. ... condition for the identity function on S to exist. ...
    (sci.math)