On the distribution of Lachlan nonsplitting bases

  title={On the distribution of Lachlan nonsplitting bases},
  author={S. Barry Cooper and Angsheng Li and Xiaoding Yi},
  journal={Archive for Mathematical Logic},
Abstract. We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base(LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there is a non-Low2 c.e. degree which bounds no Lachlan nonsplitting base. 

Splitting and nonsplitting, II: A low2 c.e. degree above which 0′ is not splittable

Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the low2-ness requirements to be satisfied, and the proof given involves new techniques with potentially wider application.


A computably enumerable (c.e.) Turing degree is a diamond base if and only if it is the bottom of a diamond of c.e. degrees with top 0′. Cooper and Li [3] showed that no low2 c.e. degree can bound a

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A given c.e. degree a > 0 has a non-trivial splitting into c.e. degrees v and w if a is the join of v and w and v | w. A Lachlan Non-Splitting Pair is a pair of c.e. degrees such that a > d and there

Definable Relations on the Computably Enumerable Degrees

This work reviews recent developments in the study of definable relations on the com­ putably enumerable (c.e.) degrees, including the following aspects: structure and hierarchies, automorphisms, and Continuity in c.e. degrees.

Dag Normann : Words at the funeral of S .

Richard Elwes School of Mathematics, University of Leeds, Leeds, LS2 9JT, England R.H.Elwes@leeds.ac.uk Andy Lewis-Pye Department of Mathematics, Columbia House, London School of Economics, Houghton

S. Barry Cooper (1943-2015)

A nonlow2 R. E. Degree with the Extension of Embeddings Properties of a low2 Degree

We construct a nonlow2 r.e. degree d such that every positive extension of embeddings property that holds below every low2 degree holds below d. Indeed, we can also guarantee the converse so that



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The proof of the main theorem involves a method that has been developed elsewhere to deal with situations in which a partial recursive functional may interfere infinitely often with an opposed requirement of lower priority.

Minimal pairs and high recursively enumerable degrees

  • S. Cooper
  • Mathematics
    Journal of Symbolic Logic
  • 1974
It is shown (uniformly) that every high r.e.degree of recursively enumerable degrees contains a high set in the sense of Robert W. Robinson [3].

Bounding minimal pairs

  • A. Lachlan
  • Mathematics
    Journal of Symbolic Logic
  • 1979
The first theorem is proved in §2 and the second in §3, which is a straightforward variation on the original minimal pair construction, but the proof of the first theorem has some novel features.

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By a degree is meant a degree of recursive unsolvability. A degree is recursively enumerable (r.e.) just if it contains a recursively enumerable subset of N, the set of non-negative integers. Two

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Recursion theory deals with computability on the natural numbers. A function ƒ from N to N is computable (or recursive) if it can be calculated by some program on a Turing machine, or equivalently on


  • R. Friedberg
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1957
If we have shown that for 1nf < jal, Vhu for sufficiently small h have their L (N2IaI +1)-norms uniformly bounded in h and since the same is true for Vh¶/i and Vh c#I, we may apply relation (3) to v

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It is proved that the recursively enumerable T‐degrees satisfy a weak form of the bi‐interpretability conjecture which implies that all jump classes Lown and Highn−1 n ⩾ 2 are definable in R without parameters.

Recursively enumerable sets and degrees

TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets