Non-distributive upper semilattice of Kleene degrees

1999 ◽  
Vol 64 (1) ◽  
pp. 147-158
Author(s):  
Hisato Muraki
Keyword(s):  
Upper Semilattice

AbstractK denotes the upper semilattice of all Kleene degrees. Under ZF + AD + DC. K is well-ordered and deg(XSJ) is the next Kleene degree above deg(X) for X ⊆ ωω (see [4] and [5, Chapter V]). While, without AD, properties of K are not always clear. In this note, we prove the non-distributivity of K under ZFC (§1), and that of Kleene degrees between deg(X) and deg(XSJ) for some X under ZFC + CH (§2.3).

An α-finite injury method of the unbounded type

1976 ◽  
Vol 41 (1) ◽  
pp. 1-17
Author(s):  
C. T. Chong
Keyword(s):  
Fine Structure ◽  
Initial Segment ◽  
Fundamental Importance ◽  
Upper Semilattice ◽  
The Past ◽  
Recursively Enumerable ◽  
Background Material ◽  
A Priority ◽  
Admissible Ordinals ◽  
Priority Argument

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα, much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

Subdirectly irreducible generalized sums of upper semilattice ordered systems of algebras

2002 ◽  
Vol 47 (2) ◽  
pp. 201-211
Author(s):  
J. Płonka ◽  
Z. Szylicka
Keyword(s):  
Subdirectly Irreducible ◽  
Upper Semilattice

Some properties of the upper semilattice of computable families of computably enumerable sets

2021 ◽  
Vol 60 (2) ◽  
pp. 195-209
Author(s):  
M. Kh. Faizrakhmanov
Keyword(s):  
Computably Enumerable Sets ◽  
Upper Semilattice ◽  
Computably Enumerable

Minimal degrees and the jump operator

1973 ◽  
Vol 38 (2) ◽  
pp. 249-271 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Minimal Degree ◽  
Partial Ordering ◽  
Jump Operator ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Original Number ◽  
Nonzero Degree ◽  
The Relationship ◽  
Degrees Of Unsolvability

The jump a′ of a degree a is defined to be the largest degree recursively enumerable in a in the upper semilattice of degrees of unsolvability. We examine below some of the ways in which the jump operation is related to the partial ordering of the degrees. Fried berg [3] showed that the equation a = x′ is solvable if and only if a ≥ 0′. Sacks [13] showed that we can find a solution of a = x′ which is ≤ 0′ (and in fact is r.e.) if and only if a ≥ 0′ and is r.e. in 0′. Spector [16] constructed a minimal degree and Sacks [13] constructed one ≤ 0′. So far the only result concerning the relationship between minimal degrees and the jump operator is one due to Yates [17] who showed that there is a minimal predecessor for each non-recursive r.e. degree, and hence that there is a minimal degree with jump 0′. In §1, we obtain an analogue of Friedberg's theorem by constructing a minimal degree solution for a = x′ whenever a ≥ 0′. We incorporate Friedberg5s original number-theoretic device with a complicated sequence of approximations to the nest of trees necessary for the construction of a minimal degree. The proof of Theorem 1 is a revision of an earlier, shorter presentation, and incorporates many additions and modifications suggested by R. Epstein. In §2, we show that any hope for a result analogous to that of Sacks on the jumps of r.e. degrees cannot be fulfilled since 0″ is not the jump of any minimal degree below 0′. We use a characterization of the degrees below 0′ with jump 0″ similar to that found for r.e. degrees with jump 0′ by R. W. Robinson [12]. Finally, in §3, we give a proof that every degree a ≤ 0′ with a′ = 0″ has a minimal predecessor. Yates [17] has already shown that every nonzero r.e. degree has a minimal predecessor, but that there is a nonzero degree ≤ 0′ with no minimal predecessor (see [18]; or for the original unrelativized result see [10] or [4]).

On finite lattices of degrees of constructibility of reals

1976 ◽  
Vol 41 (2) ◽  
pp. 313-322 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Standard Model ◽  
Finite Lattice ◽  
Partial Ordering ◽  
Turing Degrees ◽  
Upper Semilattice ◽  
Finite Lattices ◽  
Definition Of ◽  
The Universe

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ωi such that the following holds:(1) For any s, s′ Є Ui, i Є ω, k, m Є l, k ≤Km & s(m) = s′(m) → s(k) = s′(k).(2) For any s Є Ui, i Є ω, s(0) = 0 where 0 is the least element of K.(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where vK denotes the join in K.

DEFINABILITY OF THE JUMP OPERATOR IN THE ENUMERATION DEGREES

2003 ◽  
Vol 03 (02) ◽  
pp. 257-267 ◽  
Author(s):  
I. Sh. KALIMULLIN
Keyword(s):  
Jump Operator ◽  
Enumeration Degrees ◽  
Upper Semilattice

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

A survey of partial degrees

1975 ◽  
Vol 40 (2) ◽  
pp. 130-140 ◽  
Author(s):  
Leonard P. Sasso
Keyword(s):  
Equivalence Classes ◽  
Turing Machines ◽  
Systems Of Equations ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Upper Semilattice ◽  
Enumeration Reducibility ◽  
Turing Reducibility ◽  
Recursive Operators ◽  
Relative Enumerability

Partial degrees are equivalence classes of partial natural number functions under some suitable extension of relative recursiveness to partial functions. The usual definitions of relative recursiveness, equivalent in the context of total functions, are distinct when extended to partial functions. The purpose of this paper is to compare the upper semilattice structures of the resulting degrees.Relative partial recursiveness of partial functions was first introduced in Kleene [2] as an extension of the definition by means of systems of equations of relative recursiveness of total functions. Kleene's relative partial recursiveness is equivalent to the relation between the graphs of partial functions induced by Rogers' [10] relation of relative enumerability (called enumeration reducibility) between sets. The resulting degrees are hence called enumeration degrees. In [2] Davis introduces completely computable or compact functionals of partial functions and uses these to define relative partial recursiveness of partial functions. Davis' functionals are equivalent to the recursive operators introduced in Rogers [10] where a theorem of Myhill and Shepherdson is used to show that the resulting reducibility, here called weak Turing reducibility, is stronger than (i.e., implies, but is not implied by) enumeration reducibility. As in Davis [2], relative recursiveness of total functions with range ⊆{0, 1} may be defined by means of Turing machines with oracles or equivalently as the closure of initial functions under composition, primitive re-cursion, and minimalization (i.e., relative μ-recursiveness). Extending either of these definitions yields a relation between partial functions, here called Turing reducibility, which is stronger still.

The upper semilattice of degrees ≤ 0′ is complemented

1981 ◽  
Vol 46 (4) ◽  
pp. 705-713 ◽  
Author(s):  
David B. Posner
Keyword(s):  
Upper Semilattice

Let denote the set of degrees ≤ 0′. A degree a ≤ 0′ is said to be complemented in if there exists a degree b ≤ 0′ such that b ∪ a = 0′ and b ∩ a = 0. R.W. Robinson (cf. [11]) showed that every degree a ≤ 0′ satisfying a″ = 0″ is complemented in and the author [8] showed that every degree a ≤ 0′ satisfying a′ = 0″ is complemented in . Also, in [2], R. L. Epstein showed that every r.e. degree is complemented in . In this paper we will show that in fact every degree ≤ 0′ is complemented in . We will further show that the same is true in the upper semilattice of degrees ≤ c, where c is any complete degree. This is in contrast to the situation in the upper semilattice of r.e. degrees in which, as Lachlan [6] has shown, no degree other than 0 and 0′ is complemented.

Supplement to Yu. L. Ershov's article "The upper semilattice of numerations of a finite set"

1975 ◽  
Vol 14 (3) ◽  
pp. 176-178 ◽  
Author(s):  
E. A. Palyutin
Keyword(s):  
Upper Semilattice ◽  
Finite Set

Initial segments of the degrees of unsolvability Part II: minimal degrees

1970 ◽  
Vol 35 (2) ◽  
pp. 243-266 ◽  
Author(s):  
C. E. M. Yates
Keyword(s):  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Upper Semilattice ◽  
Degrees Of Unsolvability ◽  
Made In

The first of this sequence of papers [21] surveyed the progress that has been made in embedding partially ordered sets as initial segments of the uncountable upper semilattice D of degrees of unsolvability. The principal concern of the present and subsequent parts will be with initial segments of the countable upper semilattice D (≦0(1)) of degrees ≦0(1); a summary appeared in [21].

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