Introduction. One of the most important objects in the study of the recursive functions is the universal computable function, which makes it possible to compute the values of those functions in a uniform and algorithmic way. That is, if is an acceptable system of indices (see [3]) for the unary partial computable functions, then the function

is also computable.

Hence, it is interesting to study its properties, including its symmetry. The usual way to accomplish this is to study the automorphisms, isomorphic embeddings and homomorphisms of the object (see [1] for the definitions). In this case we will study the corresponding functions for the graph of ϕ. We will denote this predicate by P, so

for all natural numbers e, x and y.

A permutation f of the natural numbers will be an automorphism [1] of this predicate if

for all natural numbers e, x and y; that is, if

for all natural numbers e, x and y.

Thus, these automorphisms are changes of coordinates with respect to which the computations of the recursive functions are performed in exactly the same way as in the original model. In other words, these are the symmetric transformations which take into account the way the computable functions are actually computed.

Another interesting predicate closely related to P is the one that expresses membership of an element to an r.e. set. Since r.e. sets are exactly the domains of the unary partial computable functions (see, for instance, [4]), if we define (as usual)

the new predicate, which we will denote by W, will have the form

for x, y natural numbers.