Minimal determining sets of locally finitely-determined functionals

Let N denote the set of natural numbers, N^N the set of all total functions from N into N. By functional we mean any function whose domain is N^N. if F is a functional and A is a partial function from N into N, we say that A is a determining segment (ds) of F if F has the same value on any two total extensions of A. A ds is called minimal (mds) if it does not properly contain another ds. For α ∈ N^N, denote by F^* (α) the set of all mds's of F which are subsets of α. A functional F is called finitely-determined (fd) if every α ∈ N^N contains a finite ds. F is locally fd (lfd) if there exists a set {F_i |i ∈ I} of fd functionals such that F(α) = {F_i(α) | i ∈I} and F_i (α;) ≠ F_j (β) for i ≠ j and α, β ∈ N^N. Total continuous operators (from N^N to N^N) are lfd. Examples for fd F show that F^* (α) may contain (even 2^Nא) infinite mds's. The two main results for lfd functionals are that every ds contains a mds and that if F^*(α) consists only of finite sets then F^* (α) is itself finite. This follows from a combinatorial Theorem. If A = {n-ary union}^∞_n=1A_n where the A_n's are finite and A_n ⊄ A_m for n ≠ m, then ∃B ⊂ A such that ∀_n (A_n ⊄ B) and for an infinite sequence n₁, n₂,..., (A_ni - B) ∩ (A_nj - B) = ∅ when ≠ j. A partial recursive functional F, if undefined on α, behaves differently when fd or non-fd on α. From any oracle-machine for F we can effectively construct another which makes finitely many queries about α when F is undefined and fd on α.