Closed Classes of Ultimately Periodic Functions

Abstract

We introduce the concept of a recursively closed set and give a description of recursively closed classes generated by constants. These classes enter some partially ordered set, which “pierces” the lattice of all classes that consist of primitive recursive functions and are closed under superposition. In describing recursively closed classes generated by constants, we bring in the notion of an ultimately periodic function, which generalizes the concept of a periodic function. The main result is a theorem which holds that a recursively closed class generated by a set of $n$ constants coincides with a class of all ultimately periodic functions with periods dividing natural degrees of the number $n!$ which assume their values from that set. A consequence is obtaining a description of recursively closed classes generated by infinite sets of constants. In particular, it turns out that the recursively closed class generated by all constants coincides with the class of all ultimately periodic functions.