[GAP Forum] special bijective maps on right loops

[Akhilesh Yadav]

I asked a question on gap forum for calculating some special maps. But I
realized that it was not properly written. So I apologize to you.
I am again asking the same question:

A groupoid (S, o) with identity e is called a right loop if for every $x, y
$ in S, the equation $Xox = y$  has a unique solution in  S.
A right loop $(S, o)$ is called a right loop with unique inverses if for
each x in S there exists a unique element $x^{\prime}$ (we call inverse of
x) in S such that $x o x^{\prime} = e = x^{\prime} o x$.


Suppose that (S, o) is a right loop with unique inverses.  Then our
question is:
How to calculate bijective maps f on right loop S such that f(e)= e and
$f(x o y) = [f(x^{\prime})]^{\prime} o f(y)$ ?