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Monotone and 1–1 sets

Part of: Computability and recursion theory

Published online by Cambridge University Press:  09 April 2009

D. B. Madan  and
R. W. Robinson
D. B. Madan
Affiliation:
Department of Economic Statistics University of SydneySydney, N.S.W. 2006, Australia
R. W. Robinson
Affiliation:
Mathematics Department University of NewcastleNewcastle, N.S.W. 2308, Australia
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Abstract

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An infinite subset of ω is monotone (1–1) if every recursive function is eventually monotone on it (eventually constant on it or eventually 1–1 on it). A recursively enumerable set is co-monotone (co-1–1) just if its complement is monotone (1–1). It is shown that no implications hold among the properties of being cohesive, monotone, or 1–1, though each implies r-cohesiveness and dense immunity. However it is also shown that co-monotone and co-1–1 are equivalent, that they are properly stronger than the conjunction of r-maximality and dense simplicity, and that they do not imply maximality.

MSC classification

Secondary: 03D25: Recursively (computably) enumerable sets and degrees
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 1 , August 1982 , pp. 62 - 75
Copyright
Copyright © Australian Mathematical Society 1982

References

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