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Stable structures homogeneous for a finite binary language

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Abstract

LetL be a finite relational language andH(L) denote the class of all countable stableL-structuresM for which Th(M) admits elimination of quantifiers. ForM ∈H(L) define the rank ofM to be the maximum value of CR(p, 2), wherep is a complete 1-type over Ø and CR(p, 2) is Shelah’s complete rank. IfL has only unary and binary relation symbols there is a uniform finite bound for the rank ofM ∈H(L). This theorem confirms part of a conjecture of the first author. Intuitively it says that for eachL there is a finite bound on the complexity of the structures inH(L).

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Author notes

  1. Alistair H. Lachlan

    Present address: Department of Mathematics, Simon Fraser University, V5A 1S6, Burnaby, B.C., Canada

  2. Saharon Shelah

    Present address: Institute of Mathematics, The Hebrew University of Mathematics, Jerusalem, Israel

Authors and Affiliations

  1. Institute for Advanced Studies, The Hebrew University of Jerusalem, Jerusalem, Israel

    Alistair H. Lachlan & Saharon Shelah

Authors
  1. Alistair H. Lachlan
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  2. Saharon Shelah
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Additional information

Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death

This paper was conceived during the academic year 1980/81 while the authors were Fellows of the Institute for Advanced Studies of the Hebrew University of Jerusalem. The authors would like to thank the Institute for its generous support.

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Lachlan, A.H., Shelah, S. Stable structures homogeneous for a finite binary language. Israel J. Math. 49, 155–180 (1984). https://doi.org/10.1007/BF02760648

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  • DOI: https://doi.org/10.1007/BF02760648

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