The different behaviour of total and partial numberings with respect to the reducibility preorder is investigated. Partial numberings appear quite naturally in computability studies for topological spaces. The degrees of partial numberings form a distributive lattice which in the case of an infinite numbered set is neither complete nor contains a least element. Friedberg numberings are no longer minimal in this situation. Indeed, there is an infinite descending chain of non-equivalent Friedberg numberings below every given numbering, as well as an uncountable antichain.

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We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy.Among the many possible applications of the theory of generalized computable numberings propounded in [1], a particularly interesting and popular one is to arithmetical numberings, that is, numberings of families of arithmetical sets. When considering a family A of Σ 0 n -sets, generalized computable numberings can be characterized as follows: a numbering α of A is generalized computable if and only if the setIn what follows, such a numbering α will merely be referred to as Σ 0 n -computable. We recall that if α and β are numberings of the same family of objects, then α is said to be reducibleAn equivalence class of a numbering α (depending of course on the collection of numberings under study) will be denoted by the symbol deg(α). Since is a preordering relation, ≡ is an equivalence relation. If A is a family of Σ 0 n -sets, then ≡ partitions the set Com 0 n (A) of all Σ 0 n -computable numberings into equivalence classes, thus generating a degree structure, denoted by R 0 n (A) and called the Rogers semilattice of A.In this paper we continue research on isomorphism types of Rogers semilattices, started in [2]. We are interested in differences between elementary theories and isomorphism types at different arithmetical levels. In [3,4] it was shown that for every fixed level of the arithmetical hierarchy, there exist infinitely many families with pairwise different elementary theories for their Rogers semilattices. In [2] we established that, for every n, the isomorphism type of the Rogers semilattice of some Σ 0 n+5 -computable family B is different from the isomorphism type of the Rogers semilattice R 0 n+1 (A) of an arbitrary Σ 0 n+1 -computable family A. In this paper we improve on this result by showing that, for every n, the isomorphism type of the Rogers semilattice of any non-trivial (i.e., non one-element) Σ 0 n+4 -computable family B is different from the isomorphism type of the Rogers semilattice R 0 n+1 (A) of any Σ 0 n+1 -computable family A. For the unexplained terminology and notation relative to computability theory, our main references are in [5][6][7]. We will use the term computable function to connote completely defined computable functions. For the main concepts and notions of the theory of numberings and computable Boolean algebras, we ask the *