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UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

Published online by Cambridge University Press:  17 April 2014

URI ANDREWS ,
STEFFEN LEMPP ,
JOSEPH S. MILLER ,
KENG MENG NG ,
LUCA SAN MAURO  and
ANDREA SORBI
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706-1388, USA E-mail: andrews@math.wisc.edu
STEFFEN LEMPP
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706-1388, USA E-mail: lempp@math.wisc.edu
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706-1388, USA E-mail: jmiller@math.wisc.edu
KENG MENG NG
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL & MATHEMATICAL SCIENCES COLLEGE OF SCIENCE NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE E-mail: kmng@ntu.edu.sg
LUCA SAN MAURO
Affiliation:
SCUOLA NORMALE SUPERIORE PERFEZIONAMENTO IN DISCIPLINE FILOSOFICHE I-56126 PISA, ITALY E-mail: luca.sanmauro@sns.it
ANDREA SORBI
Affiliation:
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE E SCIENZE MATEMATICHE UNIVERSITÀ DEGLI STUDI DI SIENA, VIA ROMA, 56 I-53100 SIENA, ITALY E-mail: sorbi@unisi.it
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Abstract

We study computably enumerable equivalence relations (ceers), under thereducibility $R \le S$ if there exists a computable function f suchthat $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$, for every $x,y$. We show that the degrees of ceers under the equivalencerelation generated by $\le$ form a bounded poset that is neither a lower semilattice, noran upper semilattice, and its first-order theory is undecidable. We then studythe universal ceers. We show that 1) the uniformly effectively inseparable ceersare universal, but there are effectively inseparable ceers that are notuniversal; 2) a ceer R is universal if and only if $R\prime \le R$, where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes(answering an open question of Gao and Gerdes); and 3) both the index set of theuniversal ceers and the index set of the uniformly effectively inseparable ceersare ${\rm{\Sigma }}_3^0$-complete (the former answering an open question of Gao andGerdes).

Keywords

03D25.Computably enumerable equivalence relationseffectively inseparable sets

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 60 - 88
Copyright
Copyright © Association for Symbolic Logic 2014 

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