Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T06:06:35.107Z Has data issue: false hasContentIssue false
  • English
  • Français

HOMOGENEITY AND FIX-POINTS: GOING FORTH!

Published online by Cambridge University Press:  22 April 2015

ROGER VILLEMAIRE
ROGER VILLEMAIRE*
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITÉ DU QUÉBEC À MONTRÉAL C.P. 8888, SUCC. CENTRE-VILLE MONTRÉAL, QUÉBEC, H3C 3P8, CANADAE-mail: [email protected]: http://intra.info.uqam.ca/personnels/Members/villemaire_r
Get access Rights & Permissions [Opens in a new window]

Abstract

While the back-and-forth method has been often attributed to Cantor, it turns out that in the original proof of the characterisation of countable linear dense orders, the mapping is constructed in a single direction. Cameron has called this method Forth and has shown that it can fail to build an automorphism for some homogeneous structures. We give in this paper a characterisation of those homogeneous structures for which Forth always builds an automorphism. This generalises results by Cameron and McLeish.

Keywords

homogeneous structuresback-and-forthcategoricityfix-points
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 636 - 660
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Cameron, Peter J., Some treelike objects. The Quarterly Journal of Mathematics, vol. 38 (1987), no. 2, pp. 155–183.CrossRefGoogle Scholar
Cameron, Peter J., Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1990.Google Scholar
Cantor, Georg, Beiträge zur Begründung der transfiniten Mengenlehre, Mathematische Annalen, vol. 46 (1895), no. 4, pp. 481–512, English translation by Philip E. B. Jourdain in Contributions to the Founding of the Theory of Transfinite Numbers, 1952, Dover publications (originally Open Court, 1915), pp. 85–136 and 137–201.Google Scholar
Chang, C. C. and Jerome Keisler, H., Model theory, Studies in logic and the foundations of mathematics, vol. 73, North-Holland, Amsterdam, 1990.Google Scholar
Cousot, Patrick and Cousot, Radhia, Constructive versions of Tarski’s fixed point theorems. Pacific Journal of Mathematics, vol. 82 (1979), no. 1, pp. 43–57.Google Scholar
Hall, P., Wreath powers and characteristically simple groups. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58 (1962), pp. 170–184.Google Scholar
Hausdorff, F., Grundzüge der Mengenlehre, Verlag von Veit, Leipzig, 1914.Google Scholar
Hausdorff, Felix, Untersuchungen über Ordnungstypen IV, V, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 59 (1907), pp. 84–159, English translation in: Hausdorff on Ordered Sets, J. M. Plotkin, Editor, History of Mathematics sources volume 25, American Mathematical Society, London Mathematical Society, 2005.Google Scholar
Hodges, W., Building models by games, Dover Books on Mathematics Series, Dover Publications, 2006.Google Scholar
Huntington, Edward V., The continuum as a type of order: An exposition of the modern theory. The Annals of Mathematics, vol. 6 (1904), no. 4, pp. 151–184.Google Scholar
Knaster, B., Un théorème sur les fonctions d’ensembles. Annales de la société polonaise de mathématique, vol. 6 (1928), pp. 133–134.Google Scholar
Kueker, David, Back-and-forth arguments and infinitary logics, Infinitary logic: In memoriam Carol Karp (Kueker, David, editor), Lecture Notes in Mathematics, vol. 492, Springer, Berlin / Heidelberg, 1975, pp. 17–71.Google Scholar
Mcleish, S. J., The forth part of the back and forth map in countable homogeneous structures, this Journal, vol. 62 (1997), no. 3, pp. 873–890.Google Scholar
Plotkin, J. M., Who put the “back” in back-and-forth?, Logical methods: In honor of Anil Nerode’s sixtieth birthday (Crossley, J. N., Remmel, J. B., Shore, R. A., and Sweedler, M. E., editors), Birkhäuser, Ithaca, New York, 1993, pp. 705–712.CrossRefGoogle Scholar
Tarski, Alfred, A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, vol. 5 (1955), pp. 285–309.Google Scholar