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Hyperalgebraic primitive elements for relational algebraic and topological algebraic models

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Abstract

Using nonstandard methods, we generalize the notion of an algebraic primitive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field.

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References

  1. Albeverio, S., 1986, et. al, Nonstandard methods in stochastic analysis and mathematical physics, Orlando, Academic Press.

    Google Scholar 

  2. Ax, J., and S. Kochen, 1965, ‘Diophantine problems over local fields I’, Am. J. Math. 87, 605–630.

    Google Scholar 

  3. Ax, J., and S. Kochen, 1965, ‘Diophantine problems over local fields II. A complete set of axioms for p-adic number theory’, Am. J. Math. 87, 631–648.

    Google Scholar 

  4. Ax, J., and S. Kochen, 1966, ‘Diophantine problems over local fields III. Decidable fields’, Ann. Math. 83, 437–456.

    Google Scholar 

  5. Barwise, J., 1977, Handbook of mathematical logic, Amsterdam, New York, North-Holland.

    Google Scholar 

  6. Burris, S., and H. P. Sankapanavar, 1980, A course in universal algebra, New York, Springer-Verlag.

    Google Scholar 

  7. Fraleigh, J. B., 1989, A first course in abstract algebra, 4th Ed, Reading, Mass., Addison-Wesley Pub. Co.

    Google Scholar 

  8. Fried, M., and M. Järden, 1986, Field arithmetic, New York, Springer-Verlag, 161–182.

    Google Scholar 

  9. Flum, J., and M. Ziegler, 1980, Topological model theory, Berlin, Springer-Verlag.

    Google Scholar 

  10. Gehrke, M., M. Insall and K. Kaiser, 1990, ‘Some nonstandard methods applied to distributive lattices’, Zeitschr. f. Math. Logik und Grundlagen d. Math. 36, 123–131.

    Google Scholar 

  11. Gonshor, H., ‘Enlargements contain various kinds of completions’, In: Proceedings of the 1972 Victoria Symposium on Nonstandard Analysis, Springer Lecture Notes in Mathematics 369, 60–70.

  12. Hoskins, R. F., 1990, Standard and nonstandard analysis — fundamental theory, techniques and applications, London, Ellis Horwood Ltd.

    Google Scholar 

  13. Hurd, A. E., and P. A. Loeb, 1985, An introduction to nonstandard real analysis, Orlando, Academic Press.

    Google Scholar 

  14. Insall, (E.) M., 1987, ‘Finding generic points’, Master's Thesis, University of Houston.

  15. Insall, M., 1991, ‘Nonstandard methods and finiteness conditions in algebra’, To appear: Zeitschr. f. Math. Logik und Grundlagen d. Math. 37.

  16. Insall, M., 1992, ‘Some finiteness conditions in lattices-using nonstandard proof methods’, To appear: J. Austr. Math. Soc., Series A.

  17. Lenstra, H. W., Jr., 1992, ‘Algorithms in algebraic number theory’, Bull. of the Am. Math. Soc. 26, 211–244.

    Google Scholar 

  18. Luxemburg, W. A. J., 1969, ed., Applications of model theory to algebra, analysis and probability, New York, Holt, Rinehart and Winston.

    Google Scholar 

  19. Meisters, G. H., and J. D. Monk, 1973, ‘Construction of the reals via ultrapowers’, Rocky Mountain J. Math. 3, 141–158.

    Google Scholar 

  20. Naimpally, S. A., and B. D. Warrack, 1970, Proximity spaces, Cambridge, Cambridge University Press.

    Google Scholar 

  21. Robert, A., 1988, Nonstandard analysis, New York, John Wiley & Sons.

    Google Scholar 

  22. Robinson, A., 1966, Nonstandard analysis, (Studies in logic and the foundations of mathematics), Amsterdam, North-Holland.

    Google Scholar 

  23. Robinson, A., 1979, Selected papers of Abraham Robinson, H. J. Keisler, ed., New Haven, Yale University Press.

    Google Scholar 

  24. Robinson, A., ‘Model theory as a framework for algebra’, in Studies in Model Theory, M. D. Morley, ed. (Studies in Mathematics), The Mathematical Association of America, 134–157.

  25. Schmid, J., 1974, ‘Completing boolean algebras by nonstandard methods’, Zeitschr. f. Math. Logik und Grundlagen d. Math. 20, 47–48.

    Google Scholar 

  26. Schmid, J., 1977, ‘Nonstandard constructions for join — extensions of lattices’, Houston Journal of Mathematics 3, 423–439.

    Google Scholar 

  27. Wiçesław, W., 1988, Topological fields, New York, Marcel Dekker.

    Google Scholar 

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Insall, M. Hyperalgebraic primitive elements for relational algebraic and topological algebraic models. Stud Logica 57, 409–418 (1996). https://doi.org/10.1007/BF00370842

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