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INITIAL SEGMENTS OF THE DEGREES OF CEERS

Part of: Computability and recursion theory

Published online by Cambridge University Press:  18 February 2022

URI ANDREWS  and
ANDREA SORBI
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISONMADISON, WI53706-1388, USAE-mail:andrews@math.wisc.eduURL:http://www.math.wisc.edu/~andrews/
ANDREA SORBI
Affiliation:
DEPARTMENT OF INFORMATION ENGINEERING AND MATHEMATICS UNIVERSITY OF SIENA53100SIENA, ITALYE-mail:andrea.sorbi@unisi.itURL:http://www3.diism.unisi.it/~sorbi/
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Abstract

It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes.

Keywords

initial segmentcomputably enumerable equivalence relationceerdistributive lattice

MSC classification

Primary: 03D30: Other degrees and reducibilities
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1260 - 1282
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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