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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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References

REFERENCES

[1]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.Google Scholar
[2]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Crossley, J. N., editor), Upside Down A Book Company, Steel's Creek, Australia, 1981, pp. 26–41.Google Scholar
[3]Chisholm, J., The complexity of intrinsically r.e. subsets of existentially decidable models, this Journal, vol. 55 (1990), pp. 1213–1232.Google Scholar
[4]Downey, R. G., Goncharov, S. S., and Hirschfeldt, D. R., Degree spectra of relations on boolean algebras, Algebra and Logic, vol. 42 (2003), pp. 105–111.CrossRefGoogle Scholar
[5]Downey, R. G. and Jockusch, C. G. Jr., Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871–880.CrossRefGoogle Scholar
[6]Downey, R. G. and Knight, J. F., Orderings with α-th jump degree 0α, Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 545–552.Google Scholar
[7]Fraïssé, R., Theory of relations, North-Holland, Amsterdam, New York, 1986.Google Scholar
[8]Goncharov, S. S., Harizanov, V. S., Knight, J. F., McCoy, C., Miller, R. G., and Solomon, R., Enumerations in computable structure theory, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 219–246.CrossRefGoogle Scholar
[9]Harizanov, V. S., Uncountable degree spectra, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 255–263.CrossRefGoogle Scholar
[10]Harizanov, V. S., Pure computable model theory, Handbook of recursive mathematics (Ershov, Yu.L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), vol. 1, Elsevier, Amsterdam, 1998, pp. 3–114.Google Scholar
[11]Harizanov, V. S., Relations on computable structures, Contemporary mathematics, University of Belgrade, 2000, pp. 65–81.Google Scholar
[12]Harizanov, V. S. and Miller, R. G., Spectra of structures and relations, this Journal, vol. 72 (2007), pp. 324–348.Google Scholar
[13]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.CrossRefGoogle Scholar
[14]Hodges, W., A shorter model theory, Cambridge University Press, Cambridge, 1997.Google Scholar
[15]Jockusch, C. G. Jr. and Soare, R. I., Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.CrossRefGoogle Scholar
[16]Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153–193.CrossRefGoogle Scholar
[17]Khoussainov, B. and Shore, R. A., Effective model theory: The number of models and their complexity, Models and computability: Invited papers from Logic Colloquium '97 (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Notes Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193–240.CrossRefGoogle Scholar
[18]Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 1034–1042.Google Scholar
[19]Miller, R. G., The Δ20-spectrum of a linear order, this Journal, vol. 66 (2001), pp. 470–486.Google Scholar
[20]Moses, M. F., Relations intrinsically recursive in linear orders, Zeitschrift für Mathematische Logik und Grundtagen der Mathematik, vol. 32 (1986), pp. 467–472.CrossRefGoogle Scholar
[21]Richter, L. J., Degrees of structures, this Journal, vol. 46 (1981), pp. 723–731.Google Scholar
[22]Slaman, T., Relative to any nonrecursive set, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2117–2122.CrossRefGoogle Scholar
[23]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
[24]van der Waerden, B. L., Moderne Algebra, Springer, Berlin, 1930, English translation Algebra, (F. Blum and J. R. Schulenberger, translators) Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[25]Wehner, S., Enumerations, countable structures, and Turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131–2139.CrossRefGoogle Scholar