Gödel Numberings versus Friedberg Numberings

In [3], Rogers discussed the concept of Gödel numbering. He defined a semi-effective numbering, constructed a semi-lattice of equivalence classes of semi-effective numberings, and showed that all Gödel numberings belong to the unique maximal element of this semi-lattice. In [l], Friedberg gave a recursive enumeration without repetition of the set of partial recursive functions of a single variable. Friedberg's numbering is clearly a semi-effective numbering which is not a Gödel numbering.2 Question. How do Friedberg numberings (Definition 1 below) compare with Gödel numberings? More generally, where do Friedberg numberings fit into Rogers' semi-lattice? Definition 1. A Friedberg numbering t is a semi-effective numbering such that (1) Dr=N, (2) Ti^TjiH^j. Summary of results.3 I. If t is a Friedberg numbering then [t] is a minimal element of Rogers' semi-lattice. (Theorem 1.) II. There exists two Friedberg numberings t1 and t2 such that [t1] and [t2] are incomparable. (Theorem 2.) As a consequence of I and II we see that Rogers' semi-lattice is not a lattice (which answers a question raised in [3]).