Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Randomness and semigenericity
HTML articles powered by AMS MathViewer

by John T. Baldwin and Saharon Shelah PDF
Trans. Amer. Math. Soc. 349 (1997), 1359-1376 Request permission

Abstract:

Let $L$ contain only the equality symbol and let $L^+$ be an arbitrary finite symmetric relational language containing $L$. Suppose probabilities are defined on finite $L^+$ structures with ‘edge probability’ $n^{-\alpha }$. By $T^{\alpha }$, the almost sure theory of random $L^+$-structures we mean the collection of $L^+$-sentences which have limit probability 1. $T_{\alpha }$ denotes the theory of the generic structures for $\mathbb {K}_{\alpha }$ (the collection of finite graphs $G$ with $\delta _\alpha (G) = |G| - \alpha \cdot |\text {edges of $G$}|$ hereditarily nonnegative).

Theorem. $T^{\alpha }$, the almost sure theory of random $L^+$-structures, is the same as the theory $T_{\alpha }$ of the $\mathbb {K}_{\alpha }$-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

References
  • J.T. Baldwin and Niandong Shi. Stable generic structures. Annals of Pure and Applied Logic, 79: 1–35, 1996.
  • A. Baudisch. A new $\aleph _1$-categorical pure group. 1992.
  • E. Hrushovski. A stable $\aleph _0$-categorical pseudoplane. preprint, 1988.
  • D. W. Kueker and M. C. Laskowski, On generic structures, Notre Dame J. Formal Logic 33 (1992), no. 2, 175–183. MR 1167973, DOI 10.1305/ndjfl/1093636094
  • J. Lynch. Probabilities of sentences about very sparse random graphs. Random Structures and Algorithms, 3:33–53, 1992.
  • S. Shelah. 0-1 laws. preprint 550, 199?
  • S. Shelah. Zero-one laws with probability varying with decaying distance. Shelah 467, 199x.
  • Saharon Shelah and Joel Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc. 1 (1988), no. 1, 97–115. MR 924703, DOI 10.1090/S0894-0347-1988-0924703-8
  • F. Wagner. Relational structures and dimensions. In Automorphisms of first order structures, pages 153–181. Clarendon Press, Oxford, 1994.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03C10, 05C80
  • Retrieve articles in all journals with MSC (1991): 03C10, 05C80
Additional Information
  • Saharon Shelah
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Received by editor(s): September 7, 1994
  • Additional Notes: Partially supported by NSF grant 9308768 and a visit to Simon Fraser University.
    This is paper 528. Both authors thank Rutgers University and the Binational Science Foundation for partial support of this research.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 1359-1376
  • MSC (1991): Primary 03C10, 05C80
  • DOI: https://doi.org/10.1090/S0002-9947-97-01869-2
  • MathSciNet review: 1407480