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Computability of Fraïssé limits

Published online by Cambridge University Press:  12 March 2014

Barbara F. Csima ,
Valentina S. Harizanov ,
Russell Miller  and
Antonio Montalbán
Barbara F. Csima
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada, E-mail: csima@math.uwaterloo.ca, URL: http://www.math.uwaterloo.ca/~csima
Valentina S. Harizanov
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC 20052, USA, E-mail: harizanv@gwu.edu
Russell Miller
Affiliation:
Department of Mathematics, Queens College– CUNY, 65-30 Kissena Blvd., Flushing, NY 11367, USA PH.D. Programs in Mathematics and Computer Science, CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA, E-mail: Russell.Miller@qc.cuny.edu, URL: http://qcpages.qc.cuny.edu/~rmiller
Antonio Montalbán
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL, USA, E-mail: antonio@math.uchicago.edu, URL: http://www.math.uchicago.edu/~antonio/index.html
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Abstract

Fraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 66 - 93
Copyright
Copyright © Association for Symbolic Logic 2011

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