A jump operator on honest subrecursive degrees

It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let a boolean OR b and a boolean AND b denote respectively the join and the meet of the degrees a and b. This paper introduces a jump operator (.') on the honest elementary degrees and defines canonical degrees 0, 0', 0 ",... and low and high degrees analogous to the corresponding concepts for the Turing degrees. Among others, the following results about the structure of the honest elementary degrees are shown: There exist low degrees, and there exist degrees which are neither low nor high. Every degree above 0' is the jump of some degree, moreover, for every degree c above 0' there exist degrees a, b such that c = a boolean OR b = a' = b'. We have a' boolean OR b' less than or equal to (a boolean OR b)' and a' boolean AND b' greater than or equal to (a boolean AND b)'. The jump operator is of course monotonic, i.e, a less than or equal to b double right arrow a' less than or equal to b'. We prove that every situation compatible with a less than or equal to b double right arrow a' less than or equal to b' is realized in the structure, e.g. we have incomparable degrees a, b such that a' < b' and incomparable degrees a, b such that a' = b' etcetera, We are able to prove all these results without the traditional recursion theoretic constructions. Our proof method relies on the fact that the growth of the functions in a degree is bounded. This technique also yields a very simple proof of an old result, namely that the structure is a lattice.