Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T20:06:44.961Z Has data issue: false hasContentIssue false
  • English
  • Français

Boolean simple groups and boolean simple rings

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti
Gaisi Takeuti*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Get access Rights & Permissions [Opens in a new window]

Extract

Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V () of set theory. If we observe G outside V (), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V (). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.

The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 1 , March 1988 , pp. 160 - 173
Copyright
Copyright © Association for Symbolic Logic 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Eda, K. and Hibino, K., On Boolean powers of the group Z and (ω, ω)-weak distributivity, Journal of the Mathematical Society of Japan, vol. 36 (1984), pp. 619–628.CrossRefGoogle Scholar
[2] Eda, K., A Boolean power and a direct product of abelian groups, Tsukuba Journal of Mathematics, vol. 6 (1982), pp. 187–194.CrossRefGoogle Scholar
[3] Eda, K., On a Boolean power of a torsionfree abelian group, Journal of Algebra, vol. 82 (1983), pp. 83–94.CrossRefGoogle Scholar
[4] Lang, S., Algebra, Addison-Wesley, Reading, Massachusetts, 1971.Google Scholar
[5] Kamo, S., On the slender property of certain Boolean algebras, Journal of the Mathematical Society of Japan, vol. 38 (1986), pp. 493–500.CrossRefGoogle Scholar
[6] Mansfield, R., The theory of Boolean ultrapowers, Annals of Mathematical Logic, vol. 2 (1971), pp. 297–323.CrossRefGoogle Scholar
[7] Nishimura, H., An approach to the dimension theory of continuous geometry from the standpoint of Boolean valued analysis, Publications of the Research Institute for the Mathematical Sciences, vol. 21 (1985), pp. 181–190.CrossRefGoogle Scholar
[8] Ozawa, M., Boolean valued interpretation of Hilbert space theory, Journal of the Mathematical Society of Japan, vol. 35 (1983), pp. 609–627.CrossRefGoogle Scholar
[9] Ozawa, M., Boolean valued analysis and type I AW*-algebras, Proceedings of the Japan Academy, vol. 59 (1983), pp. 368–371.Google Scholar
[10] Ozawa, M., A classification of type I AW*-algebras and Boolean valued analysis, Journal of the Mathematical Society of Japan, vol. 36 (1984), pp. 589–608.CrossRefGoogle Scholar
[11] Ozawa, M., A transfer principle from von Neumann algebras to AW*-algebras, Journal of the London Mathematical Society, ser. 2, vol. 32 (1985), pp. 141–148.CrossRefGoogle Scholar
[12] Takeuti, G., Two applications of logic to mathematics, Publications of the Mathematical Society of Japan, no. 13, Iwanami Shoten, Tokyo, and Princeton University Press, Princeton, New Jersey, 1978.Google Scholar
[13] Takeuti, G., A transfer principle in harmonic analysis, this Journal, vol. 44 (1979), pp. 417–440.Google Scholar
[14] Takeuti, G., Boolean valued analysis, Applications of sheaves ( Proceedings, Durham, England, 1977 ; Fourman, M. P. et al., editors), Lecture Notes in Mathematics, vol. 753, Springer-Verlag, Berlin, 1979, pp. 714–731.CrossRefGoogle Scholar
[15] Takeuti, G., Boolean completion and m-convergence, Categorical aspects of topology and analysis ( Proceedings, Ottawa, 1980 ; Banaschewski, B., editor), Lecture Notes in Mathematics, vol. 915, Springer-Verlag, Berlin, 1982, pp. 333–350.CrossRefGoogle Scholar
[16] Takeuti, G., Von Neumann algebras and Boolean valued analysis, Journal of the Mathematical Society of Japan, vol. 35 (1983), pp. 1–21.CrossRefGoogle Scholar
[17] Takeuti, G., C*-algebras and Boolean valued analysis, Japanese Journal of Mathematics, vol. 9 (1983), pp. 207–245.CrossRefGoogle Scholar