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**11 - 16**of**16**### A Discrete Splitting Theorem for the

"... Introduction The following definitions are taken from [Cooper, 2000]. Definition 1.1 . Given a, b, and d, say d is splittable over a avoiding b i# if a, b < d and b ## a, then there exist d 0 , d 1 < d for which a < d 0 , d 1 , b ## d 0 and b ## d 1 , and d = d 0 # d 1 . . Further, d ..."

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Introduction The following definitions are taken from [Cooper, 2000]. Definition 1.1 . Given a, b, and d, say d is splittable over a avoiding b i# if a, b < d and b ## a, then there exist d 0 , d 1 < d for which a < d 0 , d 1 , b ## d 0 and b ## d 1 , and d = d 0 # d 1 . . Further, d is discretely splittable over a avoiding b i# each such d i is greater than or equal to a minimal cover m i of a such that m i

### There Exists A Maximal 3-C. E. Enumeration Degree

"... We construct an incomplete 3-c.e. enumeration degree which is maximal among the n-c.e. enumeration degrees for every n with 3 # n # #. Consequently the n-c.e. enumeration degrees are not dense for any such n. We show also that no low n-c.e. e-degree can be maximal among the n-c.e. e-degrees, for 2 # ..."

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We construct an incomplete 3-c.e. enumeration degree which is maximal among the n-c.e. enumeration degrees for every n with 3 # n # #. Consequently the n-c.e. enumeration degrees are not dense for any such n. We show also that no low n-c.e. e-degree can be maximal among the n-c.e. e-degrees, for 2 # n # #.

### Splitting and Nonsplitting in the Σ 0 2 Enumeration Degrees ∗

"... This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the lo-cal structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any hig ..."

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This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the lo-cal structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any high total e-degree below 0 ′ e is splittable over any low e-degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a ∅ ′′ ′-priority construction of a Π 0 1 e-degree unsplittable over a 3-c.e. e-degree below it. 1

### COUNTING THE BACK-AND-FORTH TYPES

"... Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many n-back-and-forth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some stru ..."

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Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many n-back-and-forth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some structure in K. In the former case we show that there is a countable set of infinitary Πn relations that captures all of the Πn information about the structures in K. In most cases where there are countably many n-back-and-forth equivalence classes, there is a computable description of them. We will show how to use this computable description to get a complete set of computably infinitary Πn formulas. This will allow us to completely characterize the relatively intrinsically Σ 0 n+1 relations in the computable structures of K, and to prove that no Turing degree can be coded by the (n − 1)st jump of any structure in K unless that degree is already below 0 (n−1). 1.

### Noncappable Enumeration Degrees Below ...

"... We prove that there exists a noncappable enumeration degree strictly below 0 0 e . ..."

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We prove that there exists a noncappable enumeration degree strictly below 0 0 e .

### A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES

"... Abstract. We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: In fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: If a, b are enumeration degrees, with a total, and a ..."

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Abstract. We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: In fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: If a, b are enumeration degrees, with a total, and a < b, then in the degree interval (a, b), one can embed every countable partial order. 1.