## A rigid cone in the truth-table degrees with jump (2004)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Kjos-hanssen04arigid,

author = {Bjørn Kjos-hanssen},

title = {A rigid cone in the truth-table degrees with jump},

year = {2004}

}

### OpenURL

### Abstract

Each automorphism of the truth-table degrees with order and jump is equal to the identity on a cone with base 0 (4). A degree structure is said to be rigid on a cone if each automorphism of the structure is equal to the identity on the set of degrees above a fixed degree. It is known [4] that the structure of the Turing degrees with jump is rigid on a cone. This is shown by applying a jump inversion theorem and results on initial segments. In this paper it is shown using a weaker jump inversion theorem that also the structure of truth-table degrees with jump is rigid on a cone. For definitions relating to initial segments we refer to [13]. 1 Initial segment construction Lemma 1.1. Suppose for each e, g lies on a tree Te which is e-splitting for some c for some tables with the properties of Proposition 4.9, in the sense of [5]. Then g is hyperimmune-free. Proof. For each e ∈ ω there exists e ∗ ∈ ω such that for all stages s and all oracles g, if {e ∗ } g s(x) ↓ then {e ∗ } g (x) = {e} g (x) and {e} g s(y) ↓ for all y ≤ x. If g lies on Te ∗ then it follows that {e}g is total and {e ∗ } T (σ) (x) ↓ for each σ of length x + 1. Hence {e} g = {e ∗ } g is dominated by the recursive function f(x) = max{{e} T (σ) (x) : |σ | = x + 1}. Proposition 1.2. Let L be a Σ 0 4(y)-presentable upper semilattice with least and greatest element. Then there exist t, i, g such that 1. t: ω → 2 is 0 ′ ′-computable, 2. i is the characteristic function of a set I such that I ≤m y (3), 3. g ′ ′ (e) = t(i(0),..., i(e)) for all e ∈ ω, 4. [0, g] is isomorphic to L, and 5. g is hyperimmune-free 1 Proof. The proof in [5] must be modified to employ the lattice tables of Proposition

### Citations

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Degrees of Unsolvability
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26 |
et al., A Compendium of Continuous Lattices
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On the general theory of algebraic systems
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7 |
Density of a final segment of the truth-table degrees
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Citation Context ...in Theorem Definition 3.1. In the tt-degrees we denote the order by ≤. If x, y are ttdegrees, we say that x ≡T y if for some X ∈ x and Y ∈ y, we have X ≡T Y . The following theorem is due to Mohrherr =-=[9]-=-. Theorem 3.2. Let n ≥ 1 and a ≥ 0 (n) . Then for some b, a = b (n) . Proposition 3.3. For each g, [0, g] is Σ 0 3(g)-presentable. Proof. An analysis of the definition of tt-reducibility. Lemma 3.4. E... |

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A new proof of the congruence lattice representation theorem, Algebra Universalis 6
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Citation Context .... In this paper it is shown using a weaker jump inversion theorem that also the structure of truth-table degrees with jump is rigid on a cone. For definitions relating to initial segments we refer to =-=[13]-=-. 1 Initial segment construction Lemma 1.1. Suppose for each e, g lies on a tree Te which is e-splitting for some c for some tables with the properties of Proposition 4.9, in the sense of [5]. Then g ... |

4 |
Fixed points of jump preserving automorphisms of degrees
- Solovay
- 1977
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Citation Context ...ity on a cone with base 0 (4) . A degree structure is said to be rigid on a cone if each automorphism of the structure is equal to the identity on the set of degrees above a fixed degree. It is known =-=[4]-=- that the structure of the Turing degrees with jump is rigid on a cone. This is shown by applying a jump inversion theorem and results on initial segments. In this paper it is shown using a weaker jum... |

3 |
Countable initial segments of the degrees
- Lachlan, Lebeuf
- 1976
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Citation Context ...efer to [13]. 1 Initial segment construction Lemma 1.1. Suppose for each e, g lies on a tree Te which is e-splitting for some c for some tables with the properties of Proposition 4.9, in the sense of =-=[5]-=-. Then g is hyperimmune-free. Proof. For each e ∈ ω there exists e ∗ ∈ ω such that for all stages s and all oracles g, if {e ∗ } g s(x) ↓ then {e ∗ } g (x) = {e} g (x) and {e} g s(y) ↓ for all y ≤ x. ... |