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The forth part of the back and forth map in countable homogeneous structures

Published online by Cambridge University Press:  12 March 2014

S. J. McLeish
S. J. McLeish*
Affiliation:
Queen Mary and Westfield College, University of London, London, U.K., E-mail: sj.mcleish@ulst.ac.uk
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Abstract

The model theoretic ‘back and forth’ construction of isomorphisms and automorphisms is based on the proof by Cantor that the theory of dense linear orderings without endpoints is ℵ0-categorical. However, Cantor's method is slightly different and for many other structures it yields an injection which is not surjective. The purpose here is to examine Cantor's method (here called ‘going forth’) and to determine when it works and when it fails. Partial answers to this question are found, extending those earlier given by Cameron. We also give fuller characterisations of when forth suffices for model theoretic classes such as structures containing Jordan sets for the automorphism group, and ℵ0-categorical ω-stable structures. The work is based on the author's Ph.D. thesis.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 873 - 890
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Adeleke, S. A. and Macpherson, H. D., Classification of infinite primitive Jordan permutation groups, unpublished preprint, 1993.Google Scholar
[2]Adeleke, S. A. and Neumann, P. M., Relations related to betweenness: their structure and automorphisms, in preparation.Google Scholar
[3]Adeleke, S. A. and Neumann, P. M., Primitive permutation groups with primitive Jordan sets, Journal of the London Mathematical Society, to appear, 1994.Google Scholar
[4]Cameron, P. J., Going forth, unpublished manuscript.Google Scholar
[5]Cameron, P. J., Some treelike objects, Quarterly Journal of Mathematics, Oxford, vol. 38 (1987), no. 2, pp. 155–183.CrossRefGoogle Scholar
[6]Cameron, P. J., Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, Cambridge University Press, 1990.CrossRefGoogle Scholar
[7]Cantor, G., Beiträge zur Begründung der transfiniten Mengenlehre, Mathematische Annalen, vol. 46 (1895), pp. 481–512, translated as P. E. B. Jourdain, Contributions to the founding of the theory of transfinite numbers, Open Court, 1915.CrossRefGoogle Scholar
[8]Cherlin, G., Harrington, L., and Lachlan, A. H., ℵ0-categorical, ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985).CrossRefGoogle Scholar
[9]Droste, M., Structure of partially ordered sets with transitive automorphism groups, Memoirs of the American Mathematical Society, vol. 57 (1985), no. 334.CrossRefGoogle Scholar
[10]Hall, P., Wreath powers and characteristically simple groups, Proceedings of the Cambridge Philosophical Society, vol. 58 (1962), no. 2, pp. 170–184.CrossRefGoogle Scholar
[11]Hodges, W. A., Model theory, Cambridge University Press, 1993.CrossRefGoogle Scholar
[12]Huntington, E. V., The continuum as a type of order: an exposition of the modern theory, Annals of Mathematics, vol. 6 (1904), pp. 151–184.CrossRefGoogle Scholar
[13]Macpherson, H. D., A survey of Jordan groups, Automorphisms of first order structures (Kaye, R. M. and Macpherson, H. D., editors), Oxford University Press, 1994.Google Scholar
[14]McLeish, S. J., The sufficiency of going forth for first order homogeneous structures, Ph.D. thesis, Queen Mary and Westfleld College, London University, 1993.Google Scholar
[15]Neumann, P. M., The classification of some infinite Jordan groups, Unpublished preprint, 1994.Google Scholar
[16]Pillay, A. and Steinhorn, C., Definable sets in ordered structures, I, Transactions of the American Mathematical Society, vol. 295 (1986), no. 2, pp. 565–592.CrossRefGoogle Scholar
[17]Plotkin, J. M., Who put the ‘forth’ in back and forth, Logical methods: In honor of Anil Nerode's sixtieth birthday (Crossley, J. N., Rommel, , and Shore, , editors), Birkhäuser, Boston, Massachusetts, U.S.A., 1993.Google Scholar