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A compactness property of Dedekind σ-completef-rings

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Abstract

We prove that Dedekind σ-completef-rings are boundedly countably atomic compact in the language (+, −, ·,Λ, ∨, ≤). This means that wheneverΓ is a countable set of atomic formulae with parameters from some Dedekind σ-completef-ringA every finite subsystem of which admits a solution in some fixed productK of bounded closed intervals ofA, thenΓ admits a solution inK.

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References

  1. Banaschewski, B. andNelson, E.,Equational compactness in equational classes of algebras, Algebra Universalis2 (1972), 152–165.

    Google Scholar 

  2. Bigard, A.,Keimel, K. andWolfenstein, S.,Groupes et anneaux réticulés, Lecture Notes in Mathematics No. 608, Springer Verlag, 1977.

  3. Chang, C. C. andKeisler, H. J.,Model Theory, North Holland Publishing Company, 1973.

  4. Goodearl, K. R.,Handelman, D. E. andLawrence, J. W.,Affine representations of Grothendieck groups and their applications to Rickart C *-algebras and ℵ0-continuous rings, Memoirs of the American Mathematical Society234 (1980).

  5. Goodearl, K. R.,Partially ordered abelian groups with the interpolation property, Mathematical surveys and monographs, number 20, American Mathematical Society, 1986.

  6. Jech, T.,Set Theory, Academic Press, 1978.

  7. Jech, T.,Boolean-valued models, inHandbook of Boolean Algebras (Edited by J. D. Monk), North-Holland, Vol. 3, pp. 1197–1211, 1989.

  8. Jech, T.,Boolean-linear spaces, Advances in Mathematics81 (1990), 117–197.

    Google Scholar 

  9. Koppelberg, S.,General theory of Boolean Algebras, inHandbook of Boolean Algebras, vol. 1, pp. 1–307 (Edited by J. D. Monk with R. Bonnet), Elsevier, Amsterdam, 1989.

    Google Scholar 

  10. Mycielski, J. andRyll-Nardzewski, C.,Equationally compact algebras II, Fundamenta Mathematicae61 (1968), 271–281.

    Google Scholar 

  11. Smith, K.,Commutative regular rings and Boolean-valued fields, Journal of Symbolic Logic49 (1984), 281–297.

    Google Scholar 

  12. Stone, M. H.,A general theory of spectra. II, Proceedings of the National Academy of Sciences USA27 (1941), 83–87.

    Google Scholar 

  13. Taylor, W.,Some constructions of compact algebras, Annals of Mathematical Logic3, No. 4 (1971), 395–437.

    Google Scholar 

  14. Weglorz, B.,Equationally compact algebras (I), Fundamenta Mathematicae59 (1966), 289–298.

    Google Scholar 

  15. Wehrung, F.,Infective positively ordered monoids I, Journal of Pure and Applied Algebra83 (1992), 43–82.

    Google Scholar 

  16. Wehrung, F.,Boolean universes above Boolean models, Journal of Symbolic Logic58, No. 4 (December 1993), 1219–1250.

    Google Scholar 

  17. Wehrung, F., Bounded countable atomic compactness of ordered groups, Fundamenta Mathematicae148 (1995), 101–116.

    Google Scholar 

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  1. Département de Mathématiques, Université de Caen, F-14032, Caen Cedex, France

    F. Wehrung

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  1. F. Wehrung
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Wehrung, F. A compactness property of Dedekind σ-completef-rings. Algebra Universalis 36, 511–522 (1996). https://doi.org/10.1007/BF01233921

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