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Boolean universes above Boolean models

Published online by Cambridge University Press:  12 March 2014

Friedrich Wehrung
Friedrich Wehrung*
Affiliation:
Université de Caen, Departement de Mathématiques, 14032 Caen Cedex, France, E-mail: [email protected]
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Abstract

We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are “boundedly algebraically compact” in the language (+, −, ·, ∧, ∨, ≤), and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any first-order language. The proofs can be translated into “naive set theory” in a uniform way.

Keywords

AtomsBoolean modelsfirst-order languagesconvergence in lattice-ordered ringsequational compactnessalgebraic compactness
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 4 , December 1993 , pp. 1219 - 1250
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Barwise, J., Admissible sets and structures, Springer-Verlag, Berlin and New York, 1975.CrossRefGoogle Scholar
[2]Bernau, S. J., Lateral and Dedekind-complelion of Archimedean lattice groups, Journal of the London Mathematical Society, vol. 12 (1976), part 3, pp. 320–322.CrossRefGoogle Scholar
[3]Bioard, A., Keimel, K., and Wolfenstein, S., Groupes et anneaux réticulés. Lecture Notes in Mathematics, vol. 608, Springer-Verlag, Berlin and New York, 1977.Google Scholar
[4]Birkhoff, G., Lattice theory, American Mathematical Society Colloquium Publications, vol. 25, American Mathematical Society, Providence, Rhode Island 1967.Google Scholar
[5]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[6]Fuchs, L., Infinite ahelian groups, vols. 1 and 2, Academic Press, San Diego, California, 1970 and 1973.Google Scholar
[7]Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., and Scott, D. S., A compendium of continuous lattices, Springer-Verlag, Berlin and New York, 1980.CrossRefGoogle Scholar
[8]Jakubík, J., On σ-complete lattice-ordered groups, Czechoslovak Mathematical Journal, vol. 23 (1973), no. 98, pp. 164–174.CrossRefGoogle Scholar
[9]Jakubík, J., Conditionally orthogonally complete l-groups. Mathematische Nachrichten, vol. 65 (1975), pp. 153–162.CrossRefGoogle Scholar
[10]Jech, T., The axiom of choice, North-Holland, Amsterdam, 1973.Google Scholar
[11]Jech, T., Set Theory, Academic Press, San Diego, California, 1978.Google Scholar
[12]Jech, T., Boolean-valued models, Handbook of Boolean algebras, vol. 3 (Monk, J. D., editor), North-Holland, Amsterdam, 1989, pp. 1197–1211.Google Scholar
[13]Jech, T., Boolean-linear spaces, Advances in Mathematics, vol. 81 (1990), pp. 117–197.CrossRefGoogle Scholar
[14]Kelley, J. L., General Topology, revised edition, Van Nostrand, Princeton, New Jersey, 1968.Google Scholar
[15]Shortt, R. M. and Wehrung, F., Common extensions of semigroup-valued charges, preprint.Google Scholar
[16]Weglorz, B., Equationally compact algebras (I), Fundamenta Mathematieae, vol. 59 (1966), pp. 289–298.CrossRefGoogle Scholar
[17]Wehrung, F., Injective positively ordered monoids I, Journal of Pure and Applied Algebra, vol. 83 (1992), pp. 43–82.CrossRefGoogle Scholar
[18]Wehrung, F., Restricted injectivity, transfer property, and decompositions of separative positively ordered monoids, Communications in Algebra (to appear).Google Scholar