## Local initial segments of the Turing degrees (2002)

Venue: | Bull. Symbolic Logic |

Citations: | 6 - 2 self |

### BibTeX

@ARTICLE{Kjos-hanssen02localinitial,

author = {Bjørn Kjos-hanssen},

title = {Local initial segments of the Turing degrees},

journal = {Bull. Symbolic Logic},

year = {2002},

volume = {9},

pages = {26--36}

}

### OpenURL

### Abstract

Abstract. Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated. §1. Introduction. This article concerns the algebraic study of the upper semilattice of Turing degrees. Upper semilattices of interest in this regard tend to have a least element, hence for convenience the following definition is made. Definition 1.1. A unital semilattice (usl) is a structure L = (L, ∗, e) satisfying the following equalities for all a, b, c ∈ L.

### Citations

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(Show Context)
Citation Context ...are no other restrictions than the one found by Jockusch and Posner.2 1.2 Technical presentation of results This section may be skipped by readers unfamiliar with initial segment constructions as in =-=[8]-=-. By Jockusch and Posner [4], if a principal ideal in the Turing degrees [0,a] is a lattice then a ′′ = (a ∨ 0 ′)′ and hence if a ≤ 0 ′, we have a ′′ = 0 ′′. Consequently [0,a] is Σ0 3- presentable. O... |

38 |
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Citation Context ... b1 ∪ i, so ϕ(a0 ∪ b0) = ϕ(a1 ∪ b1), as desired. In the other direction, I is nonempty since it contains 0, and if a ≤ b ∈ I then a∪b = b so ϕ(a) = ϕ(a)∗e = ϕ(a)∗ϕ(b) = ϕ(a ∪ b) = ϕ(b) = e. ⊣ Spector =-=[22]-=- found the first nontrivial usl isomorphism type of an ideal of D; namely, an ideal with two elements. This was extended step by step, to all finite lattices by Lerman [10], to all countable usls by L... |

38 |
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(Show Context)
Citation Context ...ovided abundant moral and financial support.1 Chapter 1 Introduction 1.1 Intuitive presentation of results The Turing degree of unsolvability of a set of integers A, as introduced by Kleene and Post =-=[6]-=-, consists of those sets of integers B that are, intuitively, just as noncomputable as A, i.e. if we had an oracle that could answer questions about which integers are in A then we would be able to fi... |

36 | Interpretability and definability in the recursively enumerable degrees
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Citation Context ... jumppreserving. Slaman and Woodin [21] showed that if π is an automorphism of D and x ≥ 0 ′′ then π(x) = x. Another proof of this fact using Shore’s coding with exact pair technique was presented in =-=[15]-=-. Historically, initial segments were used to obtain partial results toward the rigidity of the Turing degrees. The work of Slaman and Woodin rendered this use obsolete; however recent results may rev... |

27 |
et al., “A Compendium of Continuous Lattices
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Citation Context ...ollowing lemma is available. Lemma 3.5. The category of finite usls L with usl homomorphisms ϕ is selfdual under the contravariant functor taking L to L ∗ and ϕ to ϕ ∗ , where ϕ is the Galois adjoint =-=[2]-=- ϕ ∗ of ϕ given by ϕ(a) ≤ x ↔ a ≤ ϕ ∗ (x). The following is a partial reconstruction of the history of Pudlák’s construction. Whitman [23] showed every lattice embeds in a partition lattice Jónsson [7... |

24 |
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(Show Context)
Citation Context ...2] ϕ ∗ of ϕ given by ϕ(a) ≤ x ↔ a ≤ ϕ ∗ (x). The following is a partial reconstruction of the history of Pudlák’s construction. Whitman [23] showed every lattice embeds in a partition lattice Jónsson =-=[7]-=- simplified Whitman’s construction. Grätzer and Schmidt [4] characterized congruence lattices of algebras. Pudlák proved that Grätzer and Schmidt’s proof could be simplified using a slight modificatio... |

21 |
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(Show Context)
Citation Context ...cular, the following are equivalent:sINITIAL SEGMENTS 5 1. L is a Σ3-presentable bounded lattice. 2. There exists g < 0 ′ such that L ∼ = [0, g] and [0, g] is a lattice. Proof. By Jockusch and Posner =-=[6]-=- if [0, g] is a lattice then g is GL2, g ′′ = (g ∪ 0 ′ ) ′ , and so [0, g] is Σ2(g ∪ 0 ′ )-presentable, and since both g and 0 ′ are below x, also Σ2(x)-presentable. ⊣ The first partial results toward... |

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Citation Context ... normal subgroups of a group G is isomorphic to the congruence lattice of G. The isomorphism sends the subgroup H to the relation xEy ⇔ xy −1 ∈ H. The following observation may be credited to Mal’cev =-=[12]-=-, [13]. Lemma 3.3. The congruence lattice of an algebra is homogeneous. Proof sketch: Suppose a, b, c, d ∈ Θ and a ≡k b → c ≡k d for each k ∈ L. Then 〈c, d〉 is in the equivalence relation E generated ... |

19 |
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(Show Context)
Citation Context ...′′ = g ′ ∪ 0 ′′ = (g ∪ 0 ′ ) ∪ 0 ′′ = g∪0 ′′ ≤ x, so [0, g] is Σ 0 1(x)-presentable, and this argument relativizes to a ∈ D. And for any a ∈ D, there are no HIF degrees in the interval (a, a ′ ] (see =-=[14]-=-), so the lemma is trivial for x = a ′ . ⊣ On the other hand, there are HIF degrees g below x with [0, g] of any Σ 0 2(x)presentable isomorphism type; in fact these degrees can all be forced to be not... |

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12 |
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(Show Context)
Citation Context ... ∪ b) = ϕ(b) = e. ⊣ Spector [22] found the first nontrivial usl isomorphism type of an ideal of D; namely, an ideal with two elements. This was extended step by step, to all finite lattices by Lerman =-=[10]-=-, to all countable usls by Lachlan and Lebeuf [9] and to all size ≤ ℵ1 usls with the countable predecessor property by Abraham and Shore [1]. But Groszek and Slaman [5] showed that it is consistent wi... |

11 |
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(Show Context)
Citation Context ...r taking L to L ∗ and ϕ to ϕ ∗ , where ϕ is the Galois adjoint [2] ϕ ∗ of ϕ given by ϕ(a) ≤ x ↔ a ≤ ϕ ∗ (x). The following is a partial reconstruction of the history of Pudlák’s construction. Whitman =-=[23]-=- showed every lattice embeds in a partition lattice Jónsson [7] simplified Whitman’s construction. Grätzer and Schmidt [4] characterized congruence lattices of algebras. Pudlák proved that Grätzer and... |

10 | Defining the Turing jump
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(Show Context)
Citation Context ...ere uncountably many b < a. The jump operator a ↦→ a ′ is included in the language under consideration below. Note however that in any ideal where the jump is invariant or definable, such as D itself =-=[20]-=-, jump can be removed from the language. An initial segment of D is a subset of D which is downward closed. An ideal of D is a set I ⊆ D such that there exists an usl L and an usl homomorphism ϕ : D →... |

9 |
Initial segments of the degrees of size ℵ1
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(Show Context)
Citation Context ...as extended step by step, to all finite lattices by Lerman [10], to all countable usls by Lachlan and Lebeuf [9] and to all size ≤ ℵ1 usls with the countable predecessor property by Abraham and Shore =-=[1]-=-. But Groszek and Slaman [5] showed that it is consistent with ZFC that the same statement with 2 ℵ0 in place of ℵ1 is false. Local initial segments are initial segments of the degrees below a given d... |

9 |
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(Show Context)
Citation Context ... all finite lattices by Lerman [10], to all countable usls by Lachlan and Lebeuf [9] and to all size ≤ ℵ1 usls with the countable predecessor property by Abraham and Shore [1]. But Groszek and Slaman =-=[5]-=- showed that it is consistent with ZFC that the same statement with 2 ℵ0 in place of ℵ1 is false. Local initial segments are initial segments of the degrees below a given degree, for example a degree ... |

7 |
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(Show Context)
Citation Context ...) ′ , and so [0, g] is Σ2(g ∪ 0 ′ )-presentable, and since both g and 0 ′ are below x, also Σ2(x)-presentable. ⊣ The first partial results toward Corollary 2.5 were obtained by Spector [22] and Sacks =-=[18]-=- who found minimal degrees below 0 ′′ and 0 ′ , respectively. Next, it is natural to consider initial segments below arbitrary nonzero r.e. degrees or low initial segments. Theorem 2.6. If a ∈ D and L... |

6 |
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(Show Context)
Citation Context ... of size n, a top and a bottom. It is not known whether or not every Mn is CLFA. The class CLFA includes the finite lattices that have a homogeneous lattice table in the sense of Lerman [11]. Lachlan =-=[8]-=- makes a claim that is equivalent to saying that every CLFA is an ideal in D, but does not give the construction as Lerman [10] had already shown that every finite lattice is an ideal in D, using an i... |

6 |
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(Show Context)
Citation Context ...ng their result was to find a suitable usl representation. Instead of strengthening the already quite complicated representations of [9], a natural representation was found based on a paper of Pudlák =-=[16]-=-,s4 BJØRN KJOS-HANSSEN a paper that has been celebrated for quite different reasons in the past among lattice theorists. The proof sketched above uses the theorem of Slaman-Woodin that all automorphis... |

3 |
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(Show Context)
Citation Context ...ontrivial usl isomorphism type of an ideal of D; namely, an ideal with two elements. This was extended step by step, to all finite lattices by Lerman [10], to all countable usls by Lachlan and Lebeuf =-=[9]-=- and to all size ≤ ℵ1 usls with the countable predecessor property by Abraham and Shore [1]. But Groszek and Slaman [5] showed that it is consistent with ZFC that the same statement with 2 ℵ0 in place... |

3 |
1988], Defining jump classes in the degrees below 0
- Shore
(Show Context)
Citation Context .... ⊣ On the other hand, there are HIF degrees g below x with [0, g] of any Σ 0 2(x)presentable isomorphism type; in fact these degrees can all be forced to be not GL1. It follows from results of Shore =-=[19]-=- that if g is not GL2 then the lemma is best possible, i.e. every presentation of [0, g] has degree at least g (3) . On thes6 BJØRN KJOS-HANSSEN other hand initial segment constructions lead to GL2 de... |

2 |
of unsolvability
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- 1963
(Show Context)
Citation Context ...k such that {e} g ≡T gk. This is accomplished using an elaboration of Spector’s splitting tree technique. 2. If k �≤ m, then gk �≤T gm. This is accomplished using a diagonalization argument, see e.g. =-=[11]-=-. 3. It can be determined recursively in x and uniformly in e whether or not {e} g is total. This is accomplished using e-total trees, a well-known technique, see e.g. [11]. 4. If k ≤ m then gk ≤T gm.... |