Invertible classes

This paper considers when one can invert general recursive operators which map a class of functions F to F. In this regard, we study four different notions of inversion. We additionally consider enumeration of operators which cover all general recursive operators which map F to F in the sense that, for every general recursive operator Ψ mapping F to F, there is a general recursive operator in the enumerated sequence which behaves the same way as Ψ on F. Three different possible types of enumeration are studied.