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Lachlan NonSplitting Pairs and high computably enumerable Turing degrees

A given c.e. degree a > 0 has a nontrivial splitting into c.e. degrees v and w if a is the join of v and w and v  w. A Lachlan NonSplitting Pair is a pair of c.e. degrees < a,d > such that a > d and there is no nontrivial splitting of a into c.e. degrees w and v with w > d and v > d. Lachlan [Lachlan1976] showed that such a pair exists by proving the Lachlan NonSplitting Theorem. This theorem is remarkable for its discovery of the 0'''priority method, and became known as the `Monster' due to its significant complexity. Harrington, Shore and Slaman subsequently tried to explain Lachlan's methods in more intuitive and comprehensible terms in a number of unpublished notes. Leonhardi [Leonhardi1997] then published a short account of the Lachlan NonSplitting Theorem based on these notes and generalised the theorem in a different direction. In their work on the separation of the jump class High from the jump class Low2, Shore and Slaman [SlamanShore1993] also conjectured that every high c.e. degree strictly bounds a Lachlan NonSplitting Pair, a fact which could be used to separate the two jump classes. While this separation was eventually achieved through the notion of a Slaman Triple, the conjecture itself remained an open question. Cooper, Yi and Li [CooperLiYi2002] also defined the notion of a c.e. Robinson degree as one which does not strictly bound the base d of a Lachlan NonSplitting Pair < a,d >, and sought to understand the relationship of this notion to the High/Low Hierarchy. In this dissertation we make the following two contributions. Firstly we show that a counterexample can be found to show that the account of the Lachlan NonSplitting Theorem given by Leonhardi [Leonhardi1997] fails to satisfy its requirements. By rectifying the construction, we give a complete, correct and intuitive account of the Lachlan NonSplitting Theorem. Secondly we show that the high permitting method developed by Shore and Slaman [SlamanShore1993] can be combined with the construction of the Lachlan NonSplitting Theorem just described to prove that every high c.e. degree strictly bounds a Lachlan NonSplitting Pair. From this it follows that the existence of a Lachlan NonSplitting Pair can be used to separate the jump classes High and Low2, that the distribution of Lachlan NonSplitting Pairs with respect to these jump classes mirrors the one for Slaman Triples, and that there is no high c.e. Robinson degree.
