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R. e. presented linear orders

Published online by Cambridge University Press:  12 March 2014

Dev Kumar Roy
Dev Kumar Roy*
Affiliation:
Florida International University, Miami, Florida 33199
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Extract

This paper looks at linear orders in the following way. A preordering is given, which is linear and recursively enumerable. By performing the natural identification, one obtains a linear order for which equality is not necessarily recursive. A format similar to Metakides and Nerode's [3] is used to study these linear orders. In effective studies of linear orders thus far, the law of antisymmetry (x ≦ y ∧ y ≦ x ⇒ y) has been assumed, so that if the order relation x ≦ y is r.e. then x < y is also r.e. Here the assumption is dropped, so that x < y may not be r.e. and the equality relation may not be recursive; the possibility that equality is not recursive leads to new twists which sometimes lead to negative results.

Reported here are some interesting preliminary results with simple proofs, which are obtained if one looks at these objects with a view to doing recursion theory in the style of Metakides and Nerode. (This style, set in [3], is seen in many subsequent papers by Metakides and Nerode, Kalantari, Remmel, Retzlaff, Shore, and others, e.g. [1], [4], [6], [7], [8], [11]. In a sequel, further investigations will be reported which look at r.e. presented linear orders in this fashion and in the context of Rosenstein's comprehensive work [10].

Obviously, only countable linear orders are under consideration here. For recursion-theoretic notation and terminology see Rogers [9].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 2 , June 1983 , pp. 369 - 376
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[1]Kalantari, I., Major subspaces of recursively enumerable vector spaces, this Journal, vol. 43, no. 2 (1978), pp. 293–303.Google Scholar
[2]Lauchli, H. and Leonard, J., On the elementary theory of linear order, Fundamenta Mathematicae, vol. 59 (1966), pp. 109–116.CrossRefGoogle Scholar
[3]Metakides, G. and Nerode, A., Recursively enumerable vector spaces, Annals of Mathematical Logic, vol. 11, no. 2 (1977), pp. 147–171.CrossRefGoogle Scholar
[4]Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17, no. 3 (1979), pp, 289–320.CrossRefGoogle Scholar
[5]Myhill, J.R., Creative sets, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 97–108.CrossRefGoogle Scholar
[6]Remmel, J.B., Maximal and cohesive vector spaces, this Journal, vol. 42, no. 3 (1977), pp. 400–418.Google Scholar
[7]Remmel, J.B., Recursively enumerable boolean algebras, Annals of Mathematical Logic, vol. 14 (1978), pp. 75–107.CrossRefGoogle Scholar
[18]Retzlaff, A., Simple and hypersimple vector spaces, this Journal, vol. 43, no. 2 (1978), pp. 260–269.Google Scholar
[9]Rogers, H.J. jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[10]Rosenstein, J.G., Linear orderings, Academic Press, 1982.Google Scholar
[11]Shore, R., Controlling the dependence degree of a recursively enumerable vector space, this Journal, vol. 43, no. 1 (1978), pp. 13–22.Google Scholar