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Transactions of the American Mathematical Society

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Complete coinductive theories. II


Author: A. H. Lachlan
Journal: Trans. Amer. Math. Soc. 328 (1991), 527-562
MSC: Primary 03C45; Secondary 03C68
DOI: https://doi.org/10.1090/S0002-9947-1991-1014253-1
MathSciNet review: 1014253
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Abstract: Let $ T$ be a complete theory over a relational language which has an axiomatization by $ \exists \forall $-sentences. The properties of models of $ T$ are studied. It is shown that existential formulas are stable. A theory of forking and independence based on Boolean combinations of existential formulas in $ \exists \forall $-saturated models of $ T$ is developed for which the independence relation is shown to satisfy a very strong triviality condition. It follows that $ T$ is tree-decomposable in the sense of Baldwin and Shelah. It is also shown that if the language is finite, then $ T$ has a prime model.


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  • [1] J. T. Baldwin, Definable second-order quantifiers, Model-Theoretic Logics, Springer-Verlag, New York, 1985, pp. 445-477. MR 819543
  • [2] J. T. Baldwin and D. W. Kueker, Ramsey quantifiers and the finite cover property, Pacific J.Math. 90 (1980), 11-19. MR 599315 (83e:03054)
  • [3] J. T. Baldwin and S. Shelah, Second-order quantifiers and the complexity of theories, Notre Dame J. Formal Logic 26 (1985), 229-303. MR 796638 (87h:03053)
  • [4] G. Cherlin and A. H. Lachlan, Finitely homogeneous structures, Trans. Amer. Math. Soc. 296 (1986), 815-850. MR 846608 (88f:03023)
  • [5] V. Harnik and L. Harrington, Fundamentals of forking, Ann. Pure Appl. Logic 26 (1984), 245-286. MR 747686 (86c:03032)
  • [6] I. M. Hodkinson and H. D. Macpherson, Relational structures induced by their finite induced substructures, J. Symbolic Logic 53 (1988), 222-230. MR 929387 (89e:03047)
  • [7] E. Hrushovski, Remarks on $ {\aleph _0}$-stable $ {\aleph _0}$-categorical theories, preprint.
  • [8] A. H. Lachlan, Two conjectures on the stability of $ \omega $-categorical theories, Fund. Math. 81 (1974), 133-145. MR 0337572 (49:2341)
  • [9] -, Complete theories with only universal and existential axioms, J. Symbolic Logic 52 (1987), 698-711. MR 902985 (88k:03079)
  • [10] -, Complete coinductive theories. I, Trans. Amer. Math. Soc. 319 (1990), 209-241. MR 957082 (90i:03036)
  • [11] -, Some coinductive graphs, Arch. Math. Logik 29 (1990), 213-229. MR 1062726 (91f:03066)
  • [12] H. D. Macpherson, Graphs determined by their finite induced subgraphs, J. Combin. Theory Ser. B 41 (1986), 230-234. MR 859313 (87k:05125)
  • [13] J. Schmerl, Coinductive $ {\aleph _0}$-categorical theories, J. Symbolic Logic 55 (1990), 1130-1137. MR 1071319 (92f:03024)
  • [14] S. Shelah, Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978. MR 513226 (81a:03030)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1014253-1
Article copyright: © Copyright 1991 American Mathematical Society

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