Algebraic Properties of Rogers Semilattices of Arithmetical Numberings

Section: Introductionmentioning

confidence: 98%

Local structure of Rogers semilattices of Σn 0-computable numberings

Podzorov

2005

“…sets to numberings (up to equivalence of the numberings) is called the Lachlan operator. LEMMA 1 [11,Lemma 2.2]. For every pair A, B of non-empty c.e.…”

Section: Definitionmentioning

confidence: 97%

“…For the main concepts and notions of the theory of numberings and computable Boolean algebras, we ask the reader to consult [8,9]. The basic notions, notation, and methods bearing on arithmetical numberings and their Rogers semilattices can be found in [10,11]. For the reader's convenience, and to make our discussion more self-contained, here, we recall the definition of the Lachlan operator for numberings, and summarize some of its properties in Lemma 1 below.…”

mentioning

confidence: 99%

Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

Badaev¹,

Гончаров

Sorbi³

2006

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 *

“…For the classical case of computable families of computably enumerable sets, it was shown that there exist infinitely many families with pairwise elementarily different theories of Rogers semilattices (see [1]). Nevertheless, resent studies on Rogers semilattices of arithmetic numberings for infinite families of sets at levels greater than one in the Kleene-Mostowsky hierarchy have revealed the following: -significant differences as regards algebraic and elementary properties of the semilattices in question compared with the classical case;-homogeneity of the structure of their ideals and intervals (see [2][3][4][5]). Initially, this led to the conjecture that the Rogers semilattices for infinite families of sets of any fixed high level of the arithmetic hierarchy would have similar properties.…”

mentioning

confidence: 99%

“…-homogeneity of the structure of their ideals and intervals (see [2][3][4][5]). Initially, this led to the conjecture that the Rogers semilattices for infinite families of sets of any fixed high level of the arithmetic hierarchy would have similar properties.…”

mentioning

confidence: 99%

Elementary Theories for Rogers Semilattices

Badaev¹,

Гончаров

Sorbi³

2005

It is proved that for every level of the arithmetic hierarchy, there exist infinitely many families of sets with pairwise non-elementarily equivalent Rogers semilattices.Research in elementary theories for Rogers semilattices is one of the main problems in the theory of numberings. For the classical case of computable families of computably enumerable sets, it was shown that there exist infinitely many families with pairwise elementarily different theories of Rogers semilattices (see [1]). Nevertheless, resent studies on Rogers semilattices of arithmetic numberings for infinite families of sets at levels greater than one in the Kleene-Mostowsky hierarchy have revealed the following: -significant differences as regards algebraic and elementary properties of the semilattices in question compared with the classical case;-homogeneity of the structure of their ideals and intervals (see [2][3][4][5]). Initially, this led to the conjecture that the Rogers semilattices for infinite families of sets of any fixed high level of the arithmetic hierarchy would have similar properties. Pretty soon, however, it was found out that for every level of the hierarchy, there are at least two families of that level such that their Rogers semilattices are not elementarily equivalent (see [6]). The objective of the present paper is to prove that for every level of the arithmetic hierarchy, there exist infinitely many families with pairwise non-elementarily equivalent Rogers semilattices. We recall some necessary notation and definitions. A numbering α of a family A ⊆ Σ 0 n+1 is said to beA numbering α is reducible to a numbering β if α = β • f for some computable function f . Reducibility of numberings is a pre-order relation on Com 0 n+1 (A), which induces, in the usual way, a quotient structure R 0 n+1 (A), an upper semilattice, called the Rogers semilattice of Σ 0 n+1 -computable numberings of the family A. We refer the reader to [2,4] for details on Σ 0 n+1 -computable numberings and related topics.THEOREM. For every k ∈ N , there exist infinitely many infinite Σ 0 k+1 -computable families with pairwise elementarily different Rogers semilattices.Proof. Let k be an arbitrary natural number. We will construct a sequence {B e } e 1 of infinite Σ 0 k+1 -computable families such that Th(R 0 k+1 (B e )) = Th(R 0 k+1 (B e )) *