Abstract first order computability. II

The recent development of recursion theory has turned in part toward studying notions of computability on domains other than the natural numbers. Without any attempt for completeness, we mention the theories of Takeuti [23] (and others), Machover [15], Kripke [10], Kreisel-Sacks [9] and Platek [20] on classes or sets of ordinals, Kleene's theory of recursive functionals on objects of arbitrary finite type over the integers [6], [7] (and others) and the theories of computability on arbitrary structures of Fraisse [1], Lacombe [11], [12], Kreisel [8] (and others) and more recently Platek [20], Montague [16] and Lambert [13]. (Levy's development of a hierarchy of set-theoretic predicates in [14] is also relevant.) Some of these theories attempt to abstract the computational (or combinatorial) aspects of recursion theory while others (notably Kreisel's and Montague's) conceive of recursion theory as a branch of definability theory. Here we study computability theory on abstract (unordered) domains primarily as a tool for hierarchy theory. The term "first order" in the title indicates that we restrict ourselves to computabilities relative to given functions (i.e. objects of type-i)