Hartley Rogers in his "Theory of recursive functions and effective computability" (page 55 in the first edition) writes
"What resemblance types are also isomorphism types? A final answer to this question has not been given. The only cases presently known are the type of the empty partial functions, the type of all constant functions, and the type of all universal partial functions.^1
1 A proof that these are the only cases possible could lead to an interesting algebraic axiomatizations of the partial recursive functions."
What did he mean by "algebraic axiomatization of the partial recursive functions"?