Local structure of Rogers semilattices of Σn 0-computable numberings

Abstract

We deal in specific features of the algebraic structure of Rogers semilattices of Σ 0 n -computable numberings, for n 2. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non 0 -principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.

Arithmetical D-degrees

“…What is more, to every bounded semilattice with Σ 0 n+3 -representation, there is either a coimmune or computable Σ 0 n+1 -set generating an isomorphic principal ideal of the join semilattice of m-degrees. The last statement strengthens a result of [3]; together with the results of [4], it allows us to expand the class of semilattices which are principal ideals or segments in the Roger semilattices of arithmetical numberings. This implies, in particular, a strengthening of the results on the difference of the isomorphism types of the Roger semilattices of the arithmetical numberings of the various levels of the arithmetical hierarchies presented in [5,6].…”

“…To strengthen some of those results and to obtain new ones, we extend the operator Ψ to numberings (in the same way as it was done in [3,4,11,12,14] and in different terms in some other papers). Let ν be a numbering of an arbitrary set S and let X be a nonempty computably enumerable set.…”

“…It is clear that if a family F of Σ 0 n+1 -sets is one-element then R 0 n+1 (F ) is trivial. For n > 0, this sufficient condition is also necessary: it is known that if n > 0 and a family F having more than one element has a Σ 0 n+1 -computable numbering, then R 0 n+1 (F ) is infinite [4,14]. For n = 0, the situation is more complicated.…”

“…According to a statement mentioned in the previous section and proved in [2], there is a numbering ν ≥ ν of L that is an n-Lachlan representation of L . Theorem 1 from [3] is equivalent to the following:…”

On the definition of a Lachlan semilattice

“…It is easy to show that principal ideals in semilattices of computable enumerations of finite families, in semilattices of Σ 0 2 -computable enumerations of finite families consisting of pairwise inclusion-incomparable sets [2], and in many other semilattices are precisely Lachlan semilattices. As proved in [3] a semilattice is Lachlan if and only if it is isomorphic to the principal ideal of m-degrees generated by a hypersimple set. In the same article each Lachlan semilattice is shown to be embeddable as an initial segment and an interval into some Rogers semilattice of Σ 0 n -computable enumerations.…”