Abstract.
It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let \(\mbox{\bf a} \cup \mbox{\bf b}\) and \(\mbox{\bf a} \cap \mbox{\bf b}\) denote respectively the join and the meet of the degrees \(\mbox{\bf a}\) and \(\mbox{\bf b}\). This paper introduces a jump operator (\(\cdot'\)) on the honest elementary degrees and defines canonical degrees \(\mbox{\bf 0},\mbox{\bf 0}', \mbox{\bf 0}^{\prime \prime },\ldots\) 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 \(\mbox{\bf 0}'\) is the jump of some degree, moreover, for every degree \(\mbox{\bf c}\) above \(\mbox{\bf 0}'\) there exist degrees \(\mbox{\bf a},\mbox{\bf b}\) such that \(\mbox{\bf c}=\mbox{\bf a} \cup \mbox{\bf b} = \mbox{{\bf a}}'=\mbox{\bf b}'\). We have \(\mbox{\bf a}'\cup \mbox{\bf b}' \leq (\mbox{\bf a}\cup\mbox{\bf b})'\) and \(\mbox{\bf a}'\cap \mbox{\bf b}' \geq (\mbox{\bf a}\cap \mbox{\bf b})'\). The jump operator is of course monotonic, i.e. \(\mbox{\bf a}\leq\mbox{{\bf b}}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'\). We prove that every situation compatible with \(\mbox{\bf a}\leq\mbox{\bf b}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'\) is realized in the structure, e.g. we have incomparable degrees \(\mbox{\bf a},\mbox{\bf b}\) such that \(\mbox{\bf a}'<\mbox{\bf b}'\) and incomparable degrees \(\mbox{\bf a},\mbox{\bf b}\) such that \(\mbox{\bf a}' = \mbox{\bf b}'\) 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.
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Received October 4, 1995
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Kristiansen, L. A jump operator on honest subrecursive degrees. Arch Math Logic 37, 105–125 (1998). https://doi.org/10.1007/s001530050086
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DOI: https://doi.org/10.1007/s001530050086