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Transactions of the American Mathematical Society

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Definable sets in ordered structures. II


Authors: Julia F. Knight, Anand Pillay and Charles Steinhorn
Journal: Trans. Amer. Math. Soc. 295 (1986), 593-605
MSC: Primary 03C45; Secondary 03C40, 03C50, 06F99
DOI: https://doi.org/10.1090/S0002-9947-1986-0833698-1
MathSciNet review: 833698
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Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that any 0-minimal structure $ M$ (in which the underlying order is dense) is strongly 0-minimal (namely, every $ N$ elementarily equivalent to $ M$ is 0-minimal). It is simultaneously proved that if $ M$ is 0-minimal, then every definable set of $ n$-tuples of $ M$ has finitely many "definably connected components."


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

  • [B] G. W. Brumfiel, Partially ordered rings and semi-algebraic geometry, London Math. Soc. Lecture Note Series no. 37, Cambridge Univ. Press, London, 1979. MR 553280 (81e:55029)
  • [C] G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Automata Theory and Formal Languages, Springer-Verlag, Berlin and New York, 1975. MR 0403962 (53:7771)
  • [D] L. van den Dries, Remarks on Tarski's problem concerning $ ({\mathbf{R}}, + , \cdot ,\exp )$, Logic Colloquium '82 (Lolli, Longo and Marcja, eds.), North-Holland, Amsterdam, 1984. MR 762106 (86g:03052)
  • [M] J. Mather, Stratifications and mappings, Dynamical Systems (M. Peixoto, ed.), Academic Press, New York, 1973. MR 0368064 (51:4306)
  • [PS1] A. Pillay and C. I. Steinhorn, Definable sets in ordered structures, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 159-162. MR 741730 (86c:03033)
  • [PS2] -, Definable sets in ordered structures. I, preprint, 1984; Trans. Amer. Math. Soc. 295 (1986), 565-592. MR 833697 (88b:03050a)
  • [T] A. Tarski, A decision method for elementary algebra and geometry, 2nd rev. ed., Univ. of California Press, Berkeley and Los Angeles, 1951. MR 0044472 (13:423a)
  • [W] H. Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545-556. MR 0095844 (20:2342)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0833698-1
Keywords: 0-minimal, definably connected, cell
Article copyright: © Copyright 1986 American Mathematical Society

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