We give a straightforward computable-model-theoretic definition of a property of 0 2 sets called order-computability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any easy characterization in pure computability theory. The most striking example is the construction of two computably isomorphic c.e. sets, one of which is order-computable and the other not.
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...
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.