Algebraic families of interpretations

To each family C of interpretations corresponds an equivalence relation among program schemes, namely the equivalence of the program schemes for all interpretation of C. A family C is algebraic if any two programs are C-equivalent iff every partial finite computation of one of them is C-equivalent to some partial finite computation of the other. Our main theorem states that a family C is algebraic iff it is represented with respect to the equivalence of programs by a single interpretation (a C-Herbrand interpretation) which is algebraic (in Scott´s sense, roughly speaking). We give examples of algebraic and non algebraic families.