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Decision problem for separated distributive lattices

Published online by Cambridge University Press:  12 March 2014

Yuri Gurevich
Yuri Gurevich*
Affiliation:
Ben-Gurion University, Beer Sheva, Israel
*
University of Michigan, Ann Arbor, Michigan 48104
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Abstract

It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1 ⊆ Y2 such that Y ⊆ X1, Y2 ⊆ X2 and Y1, ⋃ Y2 = X1 ⋃ X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 1 , March 1983 , pp. 193 - 196
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[Do]Doner, J., Decidability of the weak second-order theory of two successors, Notices of the American Mathematical Society, vol. 12 (11 1965), 65T–468.Google Scholar
[GKM]Gurevich, Y., Koppelberg, S. and Monk, J. Donald, The theory of Boolean algebras with quantification over ideals is undecidable under CH (manuscript).Google Scholar
[GS1]Gurevich, Y. and Shelah, S., Monadic theory of order and topology in ZFC (to appear).Google Scholar
[GS2]Gurevich, Y. and Shelah, S., The monadic quantifier and the “next world” (to appear).Google Scholar
[GS3]Gurevich, Y. and Shelah, S., Arithmetic is not interprétable in the monadic theory of the real line (in preparation).Google Scholar
[La]Lachlan, A. H., The elementary theory of recursively enumerable sets, Duke Mathematical Journal, vol. 35 (1968), pp. 123–146.CrossRefGoogle Scholar
[Lä]Laüchli, H., A decision procedure for the weak second-order theory of linear order, Proceedings of the Logic Colloquium, Honover 1966, North-Holland, Amsterdam.Google Scholar
[Ra]Rabin, M. O., Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 149 (1969), pp. 51–3.Google Scholar
[Ru]Rubin, M., The theory of Boolean algebras with a distinguished subalgebra is undecidable, Annales Scientifiques de l'Université de Clermont, Série Mathématique, vol. 13 (1976), pp. 129–133.Google Scholar
[T & W]Thatcher, J. W. and Wright, J. B., Generalized finite automata theory with an application to a decision problem of second-order logic, Mathematical Systems Theory, vol. 2, no. 1 (1968), pp. 57–81.CrossRefGoogle Scholar