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Degrees of orderings not isomorphic to recursive linear orderings. (English) Zbl 0734.03026
The 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.

##### MSC:
 03D25 Recursively (computably) enumerable sets and degrees 03D45 Theory of numerations, effectively presented structures
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##### References:
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