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A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES

Published online by Cambridge University Press:  25 June 2014

THEODORE A. SLAMAN  and
ANDREA SORBI
THEODORE A. SLAMAN
Affiliation:
DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF CALIFORNIA, BERKELEY 719 EVANS HALL #3840 BERKELEY, CA 94720-3840 USAE-mail: slaman@math.berkeley.edu
ANDREA SORBI
Affiliation:
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE E SCIENZE MATEMATICHE UNIVERSITÀ DEGLI STUDI DI SIENA VIA ROMA 56 I-53100 SIENA, ITALYE-mail: andrea.sorbi@unisi.it
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Abstract

We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: in fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: if a,b are enumeration degrees, with a total, and a < b, then in the degree interval (a,b), one can embed every countable partial order.

Keywords

03D30Enumeration reducibilityenumeration degree
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 633 - 643
Copyright
Copyright © The Association for Symbolic Logic 2014 

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