Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Complete coinductive theories. I

Author: A. H. Lachlan
Journal: Trans. Amer. Math. Soc. 319 (1990), 209-241
MSC: Primary 03C52; Secondary 03C45, 03C50, 03C65
MathSciNet review: 957082
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 quantifier-free formulas are stable. This limited stability is used to show that in $ \exists \forall $-saturated models the elementary types of tuples are determined by their $ \exists$-types and algebraicity is determined by existential formulas. As an application, under the additional assumption that no quantifier-free formula has the FCP, the models $ \mathcal{M}$ of $ T$ are completely characterized in terms of certain 0-definable equivalence relations on cartesian powers of $ M$. This characterization yields a result similar to that of Schmerl for the case in which $ T$ is $ {\aleph _0}$-categorical.

References [Enhancements On Off] (What's this?)

  • [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] 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)
  • [6] E. Hrushovski, Remarks on $ {\aleph _0}$-stable $ {\aleph _0}$-categorical theories, preprint.
  • [7] A. H. Lachlan, Complete theories with only universal and existential axioms, J. Symbolic Logic 52 (1987), 698-711. MR 902985 (88k:03079)
  • [8] -, Complete coinductive theories. II, Trans. Amer. Math. Soc. (to appear). MR 1014253 (92f:03025)
  • [9] H. D. Macpherson, Graphs determined by their finite induced subgraphs, J. Combin. Theory Ser. B 41 (1986), 230-234. MR 859313 (87k:05125)
  • [10] J. Schmerl, Coinductive $ {\aleph _0}$-categorical theories, J. Symbolic Logic (to appear). MR 1071319 (92f:03024)
  • [11] S. Shelah, Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978. MR 513226 (81a:03030)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03C52, 03C45, 03C50, 03C65

Retrieve articles in all journals with MSC: 03C52, 03C45, 03C50, 03C65

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society