## Some special pairs of $$\Sigma_2$$ e-degrees.(English)Zbl 0926.03045

An e-degree $${\mathbf a}$$ is called low if every set in $${\mathbf a}$$ is $$\Delta_2$$. In this paper it is proved that: There exists a non-zero low non-splitting e-degree (Theorem 2.1). For every non-zero low e-degree $${\mathbf a}$$ there exists a $$\Sigma_2$$ e-degree $${\mathbf b}$$ such that $${\mathbf a}\perp {\mathbf b}$$ and for every e-degree $${\mathbf z}$$, if $${\mathbf z}<{\mathbf a}$$ and $${\mathbf z}\nless{\mathbf b}$$, then there exists $${\mathbf y}$$ such that $${\mathbf y} < {\mathbf b}$$ and $${\mathbf y}\cup{\mathbf z}={\mathbf a}$$ (Theorem 3.1). From this results follows that there exist $$\Sigma_2$$ e-degrees $${\mathbf a},{\mathbf b}$$ such that $${\mathbf a}\perp {\mathbf b}$$ and every e-degree strictly below $${\mathbf a}$$ is also below $${\mathbf b}$$.

### MSC:

 03D28 Other Turing degree structures

### Keywords:

enumeration degrees; Turing degrees; low degrees
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### References:

 [1] Some results on the structure of the 2 enumeration degrees. Ph. D. dissertation, Simon Fraser University 1989. [2] Cooper, J. Symbolic Logic 47 pp 854– (1982) [3] Cooper, J. Symbolic Logic 49 pp 503– (1984) [4] Enumeration reducibility, nondeterministic computation, and relative computability of partial functions. In: Recursion Theory Week, Oberwolfach 1989 (, and , eds.), Lecture Notes in Mathematics 1432, Springer-Verlag, Berlin-Heidelberg-Now York 1990, pp. 57–110. [5] Cooper, Zeitschrift Math. Logik Grundlagen Math. 34 pp 491– (1988) [6] Some results on enumeration reducibility. Ph. D. dissertation, Simon Fraser University 1971. [7] McKevoy, J. Symbolic Logic 50 pp 839– (1985) [8] McKevoy, J. Symbolic Logic 50 pp 983– (1985) [9] Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York 1967. · Zbl 0183.01401 [10] and , Definability in the enumeration degrees. Preprint. · Zbl 0906.03043 [11] Recursively Enumerable Sets and Degrees. Springer-Verlag, Berlin-Heidelberg-New York 1987.
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