Sufficient conditions for zero-one laws

Author:
Jason P. Bell

Journal:
Trans. Amer. Math. Soc. **354** (2002), 613-630

MSC (1991):
Primary 60F20; Secondary 05A16

DOI:
https://doi.org/10.1090/S0002-9947-01-02884-7

Published electronically:
September 28, 2001

MathSciNet review:
1862560

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Abstract | References | Similar Articles | Additional Information

Abstract: We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size is bounded above by a polynomial in , then has a monadic second order - law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most is bounded above by a polynomial in , then this class has a first order - law. These results cover all known natural examples of classes of structures that have been proved to have a logical - law by Compton's method of analyzing generating functions.

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Additional Information

**Jason P. Bell**

Affiliation:
Department of Mathematics, University of California San Diego, La Jolla, California 92093-0112

Email:
jbell@math.ucsd.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02884-7

Received by editor(s):
April 10, 2000

Received by editor(s) in revised form:
May 18, 2001

Published electronically:
September 28, 2001

Additional Notes:
I am indebted to Stan Burris for pointing out that results obtained in the additive case lift to the multiplicative case, to John Lawrence for helping with an application, and to the referee for valuable comments regarding the presentation.

Article copyright:
© Copyright 2001
American Mathematical Society