Relationally defined clones of tree functions closed under selection or primitive recursion

Abstract

We investigate classes of tree functions which are closed under composition and primitive recursion or selection (a restricted form of recursion). The main result is the characterization of those finitary relations ς (on the set of all trees of a fixed signature) for which the clone of tree functions preserving ς is closed under selection. Moreover, it turns out that such clones are closed also under primitive recursion.