Degrees of orderings not isomorphic to recursive linear orderings.

*(English)*Zbl 0734.03026The paper under review concerns the question: What degrees of unsolvability contain linear orderings which are not isomorphic to recursive linear orderings? It is known that for any degre \(\underset{\tilde{}} a\) such that \(\underset{\tilde{}} a''>\underset{\tilde{}} 0''\) there is a linear ordering of degree \(\underset{\tilde{}} a\) not isomorphic to any recursive ordering. The next natural question was raised by Julia Knight: Is every linear ordering of low degree isomorphic to a recursive linear ordering? Next, the authors prove two theorems which both give a negative answer to this question:

Theorem 1. For any nonzero r.e. degree \(\underset{\tilde{}} c\) there is a linear ordering of degree \(\underset{\tilde{}} c\) which is not isomorphic to any recursive linear ordering.

Theorem 2. There is a structure \(<A,<_ A,Inf>\) of low degree such that \(<\omega,<_ A>\) is a linear ordering not isomorphic to a recursive ordering.

Here, on A, Inf(a,b)\(\Leftrightarrow \{c:\) \(a<_ Ac<_ Ab\) or \(b<_ Ac<_ Aa\}\) is infinite.

Finally, an analogue of the recursion theorem for recursive linear orderings is refuted.

Theorem 1. For any nonzero r.e. degree \(\underset{\tilde{}} c\) there is a linear ordering of degree \(\underset{\tilde{}} c\) which is not isomorphic to any recursive linear ordering.

Theorem 2. There is a structure \(<A,<_ A,Inf>\) of low degree such that \(<\omega,<_ A>\) is a linear ordering not isomorphic to a recursive ordering.

Here, on A, Inf(a,b)\(\Leftrightarrow \{c:\) \(a<_ Ac<_ Ab\) or \(b<_ Ac<_ Aa\}\) is infinite.

Finally, an analogue of the recursion theorem for recursive linear orderings is refuted.

Reviewer: M.M.Arslanov (Kazan’)

##### MSC:

03D25 | Recursively (computably) enumerable sets and degrees |

03D45 | Theory of numerations, effectively presented structures |

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\textit{C. G. Jockusch jun.} and \textit{R. I. Soare}, Ann. Pure Appl. Logic 52, No. 1--2, 39--64 (1991; Zbl 0734.03026)

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##### References:

[1] | Ash, C.J.; Jockusch, C.G.; Knight, J.F., Jumps of orderings, Trans. amer. math. soc., 319, 573-599, (1990) · Zbl 0705.03022 |

[2] | R.G. Downey and M. Moses, Recursive linear orders with incomplete successivities Trans. Amer. Math. Soc., to appear. · Zbl 0813.03028 |

[3] | Knight, J.F., Degrees coded in jumps of orderings, J. symbolic logic, 51, 1034-1042, (1986) · Zbl 0633.03038 |

[4] | Lerman, M., On recursive linear orderings, () · Zbl 0467.03045 |

[5] | Maass, W., Characterization of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets, Trans. amer. math. soc., 279, 311-336, (1983) · Zbl 0546.03024 |

[6] | Richter, L.J., Degrees of structures, J. symbolic logic, 46, 723-731, (1981) · Zbl 0512.03024 |

[7] | Soare, R.I., Automorphisms of the lattice of recursively enumerable sets, part II: low sets, Ann. math. logic, 22, 69-107, (1982) · Zbl 0526.03022 |

[8] | Soare, R.I., Recursively enumerable sets and degrees, (1987), Springer Berlin · Zbl 0667.03030 |

[9] | Spector, C., Recursive well-orderings, J. symbolic logic, 20, 151-163, (1955) · Zbl 0067.00303 |

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