Arithmetical D-degrees

Abstract

Description is given of the isomorphism types of the principal ideals of the join semilattice of m-degrees which are generated by arithmetical sets. A result by Lachlan of 1972 on computably enumerable m-degrees is extended to the arbitrary levels of the arithmetical hierarchy. As a corollary, a characterization is given of the local isomorphism types of the Rogers semilattices of numberings of finite families, and the nontrivial Rogers semilattices of numberings which can be computed at the different levels of the arithmetical hierarchy are proved to be nonisomorphic provided that the difference between levels is more than 1.In 1972, Lachlan [1] described the semilattices that are isomorphic to the principal ideals of the join semilattice of computably enumerable m-degrees. As shown later in [2], a semilattice satisfies Lachlan's description if and only if it is a distributive join semilattice with top and bottom which has Σ 0 3 -representation. By these results, the following is true: A join semilattice is isomorphic to a principal ideal of the semilattice of computably enumerable m-degrees if and only if it is a bounded distributive semilattice admitting Σ 0 3 -representation. The main result of this paper is a generalization of the last statement which allows us to extend it to arithmetical m-degrees. It is proved that for every natural number n, a join semilattice is isomorphic to a principal ideal of the semilattice of m-degrees of Σ 0 n+1 -sets if and only if this semilattice is bounded distributive and admits Σ 0 n+3 -representation. Moreover, all principal ideals of the join semilattice of m-degrees, generated by Δ 0 n+2 -sets appear to be bounded distributive semilattices having Σ 0 n+3 -representation. 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]. § 1. Principal Ideals of the Semilattice of Arithmetical m-Degrees All basic concepts of computation theory can be found in [7]; those of lattice theory can be found in [8]; and those of enumeration theory, in [9]. We assume the reader familiar with them. In the introduction of [9], we can also find some useful facts on distributive semilattices.To denote the value of a numbering ν at x, we traditionally write νx instead of ν(x), thus omitting parentheses. Given a partial function f , by δf we denote the domain of f and by ρf , the range of f . Given a quasiordered set A = A, ≤ , the associated partially ordered set will be denoted by A = A, ≤ (thus emp...

Rogers semilattices of limitwise monotonic numberings

“…Without loss of generality, we assume that R 0 n (T ) is not one-element. Then by [32,Corollary 4], the semilattice R 0 n (T ) has an ideal isomorphic to L. Towards a contradiction, assume that the structures R 0 n (T ) and R lm (S ) are isomorphic. Since, by Remark 2.4, R lm (S ) is an ideal inside R 0 2 (S ), the semilattice R 0 2 (S ) has an ideal isomorphic to L. On the other hand, [32, Lemma 7] implies that the semilattice L (as a segment inside R 0 2 (S )) admits a 0 4 -presentation; this gives a contradiction.…”

“…Proof This result is essentially a consequence of [32, Theorem 4]. By [32, Lemma 8], there exists a bounded distributive semilattice $$\mathcal {L}$$ which has a $$\Sigma ^0_5$$‐presentation, but does not admit a $$\Sigma ^0_4$$‐presentation.…”

“…Proof This result is essentially a consequence of [32, Theorem 4]. By [32, Lemma 8], there exists a bounded distributive semilattice $$\mathcal {L}$$ which has a $$\Sigma ^0_5$$‐presentation, but does not admit a $$\Sigma ^0_4$$‐presentation. Here boundedness means that $$\mathcal {L}$$ has both a least and a greatest elements.…”

“…Without loss of generality, we assume that $$\mathcal {R}^0_n(\mathcal {T})$$ is not one‐element. Then by [32, Corollary 4], the semilattice $$\mathcal {R}^0_n(\mathcal {T})$$ has an ideal isomorphic to $$\mathcal {L}$$. Towards a contradiction, assume that the structures $$\mathcal {R}^0_n(\mathcal {T})$$ and $$\mathcal {R}_{lm}(\mathcal {S})$$ are isomorphic.…”

“…Since, by Remark 2.4, $$\mathcal {R}_{lm}(\mathcal {S})$$ is an ideal inside $$\mathcal {R}^0_2(\mathcal {S})$$, the semilattice $$\mathcal {R}^0_2(\mathcal {S})$$ has an ideal isomorphic to $$\mathcal {L}$$. On the other hand, [32, Lemma 7] implies that the semilattice $$\mathcal {L}$$ (as a segment inside $$\mathcal {R}^0_2(\mathcal {S})$$) admits a $$\Sigma ^0_4$$‐presentation; this gives a contradiction. We deduce that $$\mathcal {R}^0_n(\mathcal {T}) \not\cong \mathcal {R}_{lm}(\mathcal {S})$$.$$\Box$$…”